Returned 97 matches (100 formulae, 59 docs)
    Lookup 260.933 ms, Re-ranking 4980.698 ms
    Found 1305790 tuple postings, 157016 formulae, 20359 documents
[ formulas ] [ documents ] [ documents-by-formula ]

( x + y ) n = k = 0 n ( n k ) x n - k y k = k = 0 n ( n k ) x k y n - k .
Doc 1
0.9365, -6.0000, 29.0000, 2.4863
testing/wikipedia/v3/00133.html
( x + y ) n = k = 0 n ( n k ) x n - k y k
Doc 2
0.5554, -2.0000, 17.0000, 2.2011
testing/wikipedia/v3/00131.html
Doc 3
0.5554, -2.0000, 17.0000, 0.8380
testing/wikipedia/v3/01569.html
( y + x ) n = k = 0 n ( n k ) y n - k x k
Doc 4
0.5554, -2.0000, 13.0000, 2.6088
testing/wikipedia/v3/03031.html
( x + y ) n = i = 0 n ( n i ) x n - i y i ,
Doc 5
0.5554, -3.0000, 14.0000, 0.5554
testing/wikipedia/v3/02165.html
( x + y ) n + 1 = k = 0 n + 1 ( n + 1 k ) x n + 1 - k y k ,
Doc 1
0.9365, -6.0000, 29.0000, 2.4863
testing/wikipedia/v3/00133.html
𝐻𝑒 n ( x + y ) = k = 0 n ( n k ) x n - k 𝐻𝑒 k ( y )
Doc 6
0.4600, -8.0000, 14.0000, 1.3902
testing/wikipedia/v3/02848.html
( Δ n c ) 0 = k = 0 n ( n k ) ( - 1 ) n - k z k = ( z - 1 ) n .
Doc 7
0.4364, -13.0000, 13.0000, 1.0908
testing/wikipedia/v3/19294.html
P n ( x ) = 1 2 n k = 0 n ( n k ) 2 ( x - 1 ) n - k ( x + 1 ) k = k = 0 n ( n k ) ( - n - 1 k ) ( 1 - x 2 ) k = 2 n k = 0 n x k ( n k ) ( n + k - 1 2 n ) ,
Doc 8
0.4364, -60.0000, 15.0000, 0.8611
testing/wikipedia/v3/02190.html
1 = 1 n = ( p + q ) n = k = 0 n ( n k ) p k q n - k .
Doc 9
0.4282, -12.0000, 9.0000, 0.4282
testing/wikipedia/v3/01438.html
c n = i = 0 n x i i ! y n - i ( n - i ) ! = 1 n ! i = 0 n ( n i ) x i y n - i = ( x + y ) n n !
Doc 10
0.4108, -37.0000, 11.0000, 0.4108
testing/wikipedia/v3/06782.html
Δ n = ( E - I ) n = k = 0 n ( n k ) ( - 1 ) n - k E k ,
Doc 7
0.4364, -13.0000, 13.0000, 1.0908
testing/wikipedia/v3/19294.html
( x + y ) α = ν α ( α ν ) x ν y α - ν .
Doc 1
0.9365, -6.0000, 29.0000, 2.4863
testing/wikipedia/v3/00133.html
Doc 11
0.3964, -7.0000, 9.0000, 0.3964
testing/wikipedia/v3/05791.html
T 0 ( x ) = 1 ,   T n ( x ) = k = 1 n S ( n , k ) x k = k = 1 n { n k } x k ,  n > 0 ,
Doc 12
0.3919, -26.0000, 12.0000, 0.7247
testing/wikipedia/v3/04139.html
B n ( x ) = k = 0 n ( n k ) b n - k x k = ( b + x ) n ,
Doc 4
0.5554, -2.0000, 13.0000, 2.6088
testing/wikipedia/v3/03031.html
( 1 + 1 / n ) n = k = 0 n ( n k ) / n k = k = 0 n 1 k ! × n n × n - 1 n × × n - k + 1 n ,
Doc 13
0.3822, -35.0000, 14.0000, 0.3822
testing/wikipedia/v3/02327.html
B n ( x ) = j = 0 n ( n j ) B j x n - j
Doc 14
0.3788, -6.0000, 7.0000, 0.3788
testing/wikipedia/v3/04615.html
k = 0 n ( n k ) B n - k ( y ) x k = k = 0 n ( n k ) L ( ( 2 y ) n - k ) x k = L ( k = 0 n ( n k ) ( 2 y ) n - k x k ) = L ( ( 2 y + x ) n ) = B n ( x + y ) .
Doc 4
0.5554, -2.0000, 13.0000, 2.6088
testing/wikipedia/v3/03031.html
( x 1 + + x m ) n = | α | = n ( n α ) x α
Doc 15
0.3646, -8.0000, 9.0000, 0.3646
testing/wikipedia/v3/05760.html
s n ( x + y ) = k = 0 n ( n k ) x k s n - k ( y ) .
Doc 16
0.3646, -12.0000, 11.0000, 1.2353
testing/wikipedia/v3/03081.html
1 e k = x k n ( k - x ) ! = k = 0 n ( n k ) B k x n - k
Doc 17
0.3646, -19.0000, 9.0000, 0.3646
testing/wikipedia/v3/09424.html
B n ( x ) = k = 0 n ( n k ) B n - k x k applying the definition of Bernoulli polynomials = k = 0 n ( n k ) L ( y n - k ) x k applying the above definition = L ( k = 0 n ( n k ) y n - k x k ) since L is linear = L ( ( y + x ) n ) .
Doc 4
0.5554, -2.0000, 13.0000, 2.6088
testing/wikipedia/v3/03031.html
( 1 + x ) n = k = 0 n ( n k ) x k .
Doc 18
0.3596, -5.0000, 10.0000, 0.3596
testing/wikipedia/v3/03830.html
Δ n [ f ] ( x ) = k = 0 n ( n k ) ( - 1 ) n - k f ( x + k )
Doc 19
0.3596, -13.0000, 11.0000, 0.3596
testing/wikipedia/v3/12635.html
u n = k = 0 n ( n k ) a k ( - c ) n - k b k
Doc 20
0.3468, -9.0000, 9.0000, 0.7163
testing/wikipedia/v3/12660.html
( x y ) n = j = 1 n - 1 ( n j ) x j y n - j
Doc 21
0.3468, -10.0000, 10.0000, 0.6796
testing/wikipedia/v3/05621.html
( f g ) ( n ) = k = 0 n ( n k ) f ( k ) g ( n - k )
Doc 22
0.3468, -11.0000, 7.0000, 0.3468
testing/wikipedia/v3/07606.html
H n ( x + y ) = k = 0 n ( n k ) H k ( x ) ( 2 y ) ( n - k ) = 2 - n 2 k = 0 n ( n k ) H n - k ( x 2 ) H k ( y 2 ) .
Doc 6
0.4600, -8.0000, 14.0000, 1.3902
testing/wikipedia/v3/02848.html
( x 1 + y 1 ) n 1 ( x d + y d ) n d = k 1 = 0 n 1 k d = 0 n d ( n 1 k 1 ) x 1 k 1 y 1 n 1 - k 1 ( n d k d ) x d k d y d n d - k d .
Doc 1
0.9365, -6.0000, 29.0000, 2.4863
testing/wikipedia/v3/00133.html
( s + t ) n = k = 0 n ( n k ) ( s ) k ( t ) n - k
Doc 23
0.3328, -11.0000, 8.0000, 0.3328
testing/wikipedia/v3/06981.html
B n ( x ) = k = 0 n ( n n - k ) b k x n - k ,
Doc 24
0.3328, -11.0000, 8.0000, 0.3328
testing/wikipedia/v3/28068.html
( x + y ) n = k = 0 n ( n k ) ( x ) n - k ( y ) k ,
Doc 26
0.3328, -12.0000, 10.0000, 0.3328
testing/wikipedia/v3/01177.html
E n ( x + y ) = k = 0 n ( n k ) E k ( x ) y n - k
Doc 25
0.3328, -12.0000, 10.0000, 1.4357
testing/wikipedia/v3/03168.html
B n ( x + y ) = k = 0 n ( n k ) B k ( x ) y n - k
Doc 25
0.3328, -12.0000, 10.0000, 1.4357
testing/wikipedia/v3/03168.html
B n ( y + x ) = k = 0 n ( n k ) B n - k ( y ) x k
Doc 4
0.5554, -2.0000, 13.0000, 2.6088
testing/wikipedia/v3/03031.html
p n ( x + y ) = k = 0 n ( n k ) p k ( x ) y n - k .
Doc 27
0.3328, -13.0000, 10.0000, 0.8651
testing/wikipedia/v3/09031.html
B n ( y + x ) = k = 0 n ( n k ) B n - k ( y ) x k .
Doc 4
0.5554, -2.0000, 13.0000, 2.6088
testing/wikipedia/v3/03031.html
p n ( x + y ) = k = 0 n ( n k ) p k ( x ) p n - k ( y )
Doc 21
0.3468, -10.0000, 10.0000, 0.6796
testing/wikipedia/v3/05621.html
s n ( x + y ) = k = 0 n ( n k ) p k ( x ) s n - k ( y ) .
Doc 16
0.3646, -12.0000, 11.0000, 1.2353
testing/wikipedia/v3/03081.html
p n ( x + y ) = k = 0 n ( n k ) p k ( x ) p n - k ( y ) .
Doc 16
0.3646, -12.0000, 11.0000, 1.2353
testing/wikipedia/v3/03081.html
Doc 28
0.3328, -15.0000, 10.0000, 0.6020
testing/wikipedia/v3/02880.html
T n ( λ + μ ) = k = 0 n ( n k ) T k ( λ ) T n - k ( μ ) .
Doc 12
0.3919, -26.0000, 12.0000, 0.7247
testing/wikipedia/v3/04139.html
( x ) ( n ) = x ( x + 1 ) ( x + n - 1 ) = k = 0 n [ n k ] x k
Doc 29
0.3328, -17.0000, 9.0000, 0.3328
testing/wikipedia/v3/10245.html
𝐻𝑒 n [ α + β ] ( x + y ) = k = 0 n ( n k ) 𝐻𝑒 k [ α ] ( x ) 𝐻𝑒 n - k [ β ] ( y ) .
Doc 6
0.4600, -8.0000, 14.0000, 1.3902
testing/wikipedia/v3/02848.html
p n ( x ) = k = 0 n ( n k ) c k x n - k ;
Doc 27
0.3328, -13.0000, 10.0000, 0.8651
testing/wikipedia/v3/09031.html
ν = 0 n b ν , n ( x ) = ν = 0 n ( n ν ) x ν ( 1 - x ) n - ν = ( x + ( 1 - x ) ) n = 1.
Doc 30
0.3273, -28.0000, 10.0000, 0.3273
testing/wikipedia/v3/04108.html
( 1 + x ) n = k = 0 n ( n k ) x k
Doc 31
0.3148, -5.0000, 10.0000, 0.4343
testing/wikipedia/v3/02183.html
( 1 + x ) n = k = 0 n ( n k ) x k .
Doc 1
0.9365, -6.0000, 29.0000, 2.4863
testing/wikipedia/v3/00133.html
( 1 + x + 1 / x ) n = k = - n n ( n k ) 2 x k
Doc 32
0.3148, -11.0000, 10.0000, 0.7830
testing/wikipedia/v3/30027.html
( x 1 ) n = k 1 = n ( n k 1 ) x 1 k 1 ;  k 1 , n 0
Doc 33
0.3148, -15.0000, 9.0000, 0.3148
testing/wikipedia/v3/10096.html
μ n ( t + s ) = k = 0 n ( n k ) μ k ( t ) μ n - k ( s ) .
Doc 34
0.3148, -15.0000, 8.0000, 0.3148
testing/wikipedia/v3/07298.html
d n f = k = 0 n ( n k ) n f x k y n - k ( d x ) k ( d y ) n - k ,
Doc 35
0.3148, -24.0000, 7.0000, 0.3148
testing/wikipedia/v3/24576.html
Δ ( X n ) = k = 0 n ( n k ) X k X n - k ,
Doc 36
0.3010, -11.0000, 7.0000, 0.3010
testing/wikipedia/v3/03879.html
cos n x = k = 0 n ( n k ) cos k x sin n - k x cos ( 1 2 ( n - k ) π ) .
Doc 37
0.3010, -21.0000, 6.0000, 0.3010
testing/wikipedia/v3/01767.html
( 1 + x + x 2 ) n = j = 0 2 n ( n j - n ) 2 x j = k = - n n ( n k ) 2 x n + k
Doc 32
0.3148, -11.0000, 10.0000, 0.7830
testing/wikipedia/v3/30027.html
d n d x n [ f ( x ) g ( x ) ] = k = 0 n ( n k ) d n - k d x n - k f ( x ) d k d x k g ( x )
Doc 38
0.2948, -33.0000, 6.0000, 0.2948
testing/wikipedia/v3/16716.html
B n ( x ) = k = 0 n ( n k ) b n - k x k ,
Doc 25
0.3328, -12.0000, 10.0000, 1.4357
testing/wikipedia/v3/03168.html
( 1 + x ) n = k = 0 ( n k ) x k | x | < 1
Doc 3
0.5554, -2.0000, 17.0000, 0.8380
testing/wikipedia/v3/01569.html
Δ n a 0 = k = 0 n ( - 1 ) k ( n k ) a n - k .
Doc 39
0.2827, -12.0000, 5.0000, 0.2827
testing/wikipedia/v3/15969.html
L n ( x ) = k = 0 n ( n k ) ( - 1 ) k k ! x k .
Doc 40
0.2827, -13.0000, 9.0000, 0.2827
testing/wikipedia/v3/07059.html
( x ) m ( x ) n = k = 0 m ( m k ) ( n k ) k ! ( x ) m + n - k .
Doc 41
0.2827, -18.0000, 6.0000, 0.2827
testing/wikipedia/v3/03187.html
( u v ) ( n ) ( x ) = k = 0 n ( n k ) u ( n - k ) ( x ) v ( k ) ( x ) .
Doc 42
0.2827, -21.0000, 8.0000, 0.2827
testing/wikipedia/v3/03398.html
k = 0 n ( n k ) A k B n - k .
Doc 43
0.2692, -6.0000, 6.0000, 0.2692
testing/wikipedia/v3/07188.html
c n = k = 0 n ( n k ) a k b n - k
Doc 28
0.3328, -15.0000, 10.0000, 0.6020
testing/wikipedia/v3/02880.html
n , k ( n k ) x k y n = 1 1 - y - x y .
Doc 2
0.5554, -2.0000, 17.0000, 2.2011
testing/wikipedia/v3/00131.html
n , k ( n + k k ) x k y n = 1 1 - x - y .
Doc 2
0.5554, -2.0000, 17.0000, 2.2011
testing/wikipedia/v3/00131.html
Δ n x m = k = 0 n ( - 1 ) n - k ( n k ) ( x + k ) m
Doc 25
0.3328, -12.0000, 10.0000, 1.4357
testing/wikipedia/v3/03168.html
n , k ( n k ) x k y n = 1 1 - ( 1 + x ) y = 1 1 - y - x y .
Doc 44
0.2692, -22.0000, 7.0000, 0.2692
testing/wikipedia/v3/02490.html
< L 1 L 2 | x n > = k = 0 n ( n k ) < L 1 | x k > < L 2 | x n - k > .
Doc 4
0.5554, -2.0000, 13.0000, 2.6088
testing/wikipedia/v3/03031.html
k = 0 n ( n k ) k n - k
Doc 45
0.2505, -3.0000, 8.0000, 0.2505
testing/wikipedia/v3/00526.html
b n = k = 0 n ( n k ) 2 ( n + k k ) 2 .
Doc 46
0.2505, -10.0000, 6.0000, 0.4034
testing/wikipedia/v3/09155.html
( N k ) = n = k m S n j = k n ( n j ) ( - 1 ) n - j .
Doc 7
0.4364, -13.0000, 13.0000, 1.0908
testing/wikipedia/v3/19294.html
k = 0 n ( n k ) 𝐻𝑒 k [ α ] ( x ) 𝐻𝑒 n - k [ - α ] ( y ) = 𝐻𝑒 n [ 0 ] ( x + y ) = ( x + y ) n .
Doc 6
0.4600, -8.0000, 14.0000, 1.3902
testing/wikipedia/v3/02848.html
P n ( x ) = k = 0 n ( - 1 ) k ( n k ) 2 ( 1 + x 2 ) n - k ( 1 - x 2 ) k .
Doc 8
0.4364, -60.0000, 15.0000, 0.8611
testing/wikipedia/v3/02190.html
D ( n ) = k = 0 n ( n k ) ( n + k k ) .
Doc 47
0.2373, -10.0000, 7.0000, 0.2373
testing/wikipedia/v3/21377.html
H n ( x ) = k = 0 n ( n + k n - k ) ( - x ) k .
Doc 48
0.2373, -14.0000, 7.0000, 0.2373
testing/wikipedia/v3/03051.html
M n ( x , β , γ ) = k = 0 n ( - 1 ) k ( n k ) ( x k ) k ! ( x - β ) n - k γ - k
Doc 49
0.2293, -27.0000, 4.0000, 0.2293
testing/wikipedia/v3/22994.html
x n = E n ( x ) + 1 2 k = 0 n - 1 ( n k ) E k ( x ) .
Doc 25
0.3328, -12.0000, 10.0000, 1.4357
testing/wikipedia/v3/03168.html
n = k ( n k ) y n = y k ( 1 - y ) k + 1 .
Doc 2
0.5554, -2.0000, 17.0000, 2.2011
testing/wikipedia/v3/00131.html
( x ) n = k = 1 n ( - 1 ) n - k L ( n , k ) x ( k ) .
Doc 50
0.2054, -17.0000, 5.0000, 0.2054
testing/wikipedia/v3/02101.html
n , k 1 ( n + k ) ! ( n + k k ) x k y n = e x + y .
Doc 2
0.5554, -2.0000, 17.0000, 2.2011
testing/wikipedia/v3/00131.html
p n ( x ) = k = 0 n a n , k x k and q n ( x ) = k = 0 n b n , k x k .
Doc 16
0.3646, -12.0000, 11.0000, 1.2353
testing/wikipedia/v3/03081.html
Doc 27
0.3328, -13.0000, 10.0000, 0.8651
testing/wikipedia/v3/09031.html
R m , n ( x ) = k = 0 min ( m , n ) ( m k ) ( n k ) k ! x k = k = 0 min ( m , n ) n ! m ! k ! ( n - k ) ! ( m - k ) ! x k .
Doc 51
0.2050, -47.0000, 7.0000, 0.2050
testing/wikipedia/v3/18569.html
𝔍 k ( a ) n = b n = i = 0 n ( - k ) n - i ( n i ) a i .
Doc 20
0.3468, -9.0000, 9.0000, 0.7163
testing/wikipedia/v3/12660.html
( 1 + x ) k = 0 n ( α k ) x k = k = 0 n ( α + 1 k ) x k + ( α n ) x n + 1 ,
Doc 52
0.1961, -28.0000, 7.0000, 0.1961
testing/wikipedia/v3/06249.html
P n ~ ( x ) = ( - 1 ) n k = 0 n ( n k ) ( n + k k ) ( - x ) k .
Doc 8
0.4364, -60.0000, 15.0000, 0.8611
testing/wikipedia/v3/02190.html
k ( n k ) x k = ( 1 + x ) n .
Doc 2
0.5554, -2.0000, 17.0000, 2.2011
testing/wikipedia/v3/00131.html
k = 0 n ( n k ) ( n n - k ) = ( 2 n n ) .
Doc 2
0.5554, -2.0000, 17.0000, 2.2011
testing/wikipedia/v3/00131.html
k = q n ( n k ) ( k q ) = 2 n - q ( n q )
Doc 2
0.5554, -2.0000, 17.0000, 2.2011
testing/wikipedia/v3/00131.html
k = d n ( n k ) ( k d ) = 2 n - d ( n d )
Doc 53
0.1734, -13.0000, 5.0000, 0.1734
testing/wikipedia/v3/03752.html
t n = k = 0 n ( - 1 ) n - k ( n k ) a k
Doc 20
0.3468, -9.0000, 9.0000, 0.7163
testing/wikipedia/v3/12660.html
a n = k = 1 n ( - 1 ) k - 1 ( n k ) 2 k ( n - k ) a n - k .
Doc 54
0.1734, -22.0000, 3.0000, 0.1734
testing/wikipedia/v3/02934.html
( n 0 ) 2 = k = 0 n n ( n - 1 ) ( n - 2 k + 1 ) ( k ! ) 2 = k = 0 n ( n 2 k ) ( 2 k k ) .
Doc 32
0.3148, -11.0000, 10.0000, 0.7830
testing/wikipedia/v3/30027.html
a n = k = 0 n c n , k ( n k ) 2 ( n + k k ) 2
Doc 46
0.2505, -10.0000, 6.0000, 0.4034
testing/wikipedia/v3/09155.html
f X ( x ; n ) = 1 2 ( n - 1 ) ! k = 0 n ( - 1 ) k ( n k ) ( x - k ) n - 1 sgn ( x - k )
Doc 55
0.1529, -33.0000, 4.0000, 0.1529
testing/wikipedia/v3/21554.html
θ n ( x ) = x n y n ( 1 / x ) = k = 0 n ( 2 n - k ) ! ( n - k ) ! k ! x k 2 n - k
Doc 56
0.1529, -34.0000, 4.0000, 0.1529
testing/wikipedia/v3/14766.html
f X ( x ; n ) = n 2 ( n - 1 ) ! k = 0 n ( - 1 ) k ( n k ) ( n x - k ) n - 1 sgn ( n x - k )
Doc 57
0.1529, -35.0000, 4.0000, 0.1529
testing/wikipedia/v3/26465.html
C n = 2 ( n 2 ) - 1 n k = 1 n - 1 k ( n k ) 2 ( n - k 2 ) C k .
Doc 58
0.1277, -25.0000, 4.0000, 0.1277
testing/wikipedia/v3/16562.html
( x y ) n = x n y n
Doc 31
0.3148, -5.0000, 10.0000, 0.4343
testing/wikipedia/v3/02183.html
( x y ) n = x n y n [ y , x ] ( n 2 ) .
Doc 59
0.1195, -12.0000, 5.0000, 0.1195
testing/wikipedia/v3/00228.html
( n h ) ( n - h k ) = ( n k ) ( n - k h )
Doc 2
0.5554, -2.0000, 17.0000, 2.2011
testing/wikipedia/v3/00131.html
( n k 1 , k 2 ) = ( n k 1 , n - k 1 ) = ( n k 1 ) = ( n k 2 ) .
Doc 2
0.5554, -2.0000, 17.0000, 2.2011
testing/wikipedia/v3/00131.html