Returned 98 matches (100 formulae, 64 docs)
    Lookup 428.448 ms, Re-ranking 4662.009 ms
    Found 1532324 tuple postings, 232607 formulae, 23170 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, 2.8460
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, 1.6288
testing/wikipedia/v3/00131.html
Doc 3
0.5554, -2.0000, 17.0000, 1.0885
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, 1.6032
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, 0.9150
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, 2.8460
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, 1.0433
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, 1.5938
testing/wikipedia/v3/19294.html
( a + b ) p = n = 0 p ( p n ) a n b p - n .
Doc 8
0.4282, -7.0000, 5.0000, 0.4282
testing/wikipedia/v3/08319.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, 1.1715
testing/wikipedia/v3/01438.html
B k ( j ) = n = 0 k ( k n ) B n j k - n .
Doc 10
0.4108, -6.0000, 5.0000, 0.4108
testing/wikipedia/v3/00144.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 11
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, 1.5938
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, 2.8460
testing/wikipedia/v3/00133.html
Doc 12
0.3964, -7.0000, 9.0000, 0.6656
testing/wikipedia/v3/05791.html
( p + q ) n = k = 0 ( n k ) p k q n - k ,
Doc 9
0.4282, -12.0000, 9.0000, 1.1715
testing/wikipedia/v3/01438.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 13
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, 1.6032
testing/wikipedia/v3/03031.html
Δ n = ( E - I ) n = j = 0 n ( n j ) ( - 1 ) n - j E j ,   n 0 .
Doc 7
0.4364, -13.0000, 13.0000, 1.5938
testing/wikipedia/v3/19294.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 14
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 15
0.3788, -6.0000, 7.0000, 0.3788
testing/wikipedia/v3/04615.html
( a X + b Y ) n = i = 0 n ( n i ) a i b n - i X i Y n - i
Doc 16
0.3788, -18.0000, 8.0000, 0.3788
testing/wikipedia/v3/15089.html
( 1 - z ) w = n 0 ( w n ) ( - 1 ) n z n = n 0 ( - 1 ) n n ! s n ( w ) z n .
Doc 17
0.3714, -24.0000, 6.0000, 0.3714
testing/wikipedia/v3/13655.html
j = k n ( n j ) ( - 1 ) n - j = - j = 0 k - 1 ( n j ) ( - 1 ) n - j = ( - 1 ) n - k ( n - 1 k - 1 ) .
Doc 7
0.4364, -13.0000, 13.0000, 1.5938
testing/wikipedia/v3/19294.html
( x 1 + + x m ) n = | α | = n ( n α ) x α
Doc 18
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 19
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 20
0.3646, -19.0000, 9.0000, 0.3646
testing/wikipedia/v3/09424.html
( 1 + x ) n = k = 0 n ( n k ) x k .
Doc 21
0.3596, -5.0000, 10.0000, 0.5778
testing/wikipedia/v3/03830.html
Δ n [ f ] ( x ) = k = 0 n ( n k ) ( - 1 ) n - k f ( x + k )
Doc 22
0.3596, -13.0000, 11.0000, 0.3596
testing/wikipedia/v3/12635.html
S = i = n 2 + 1 n ( n i ) p i ( 1 - p ) n - i .
Doc 23
0.3596, -13.0000, 8.0000, 0.3596
testing/wikipedia/v3/06439.html
( x 1 + x 2 ) n = k 1 + k 2 = n ( n k 1 , k 2 ) x 1 k 1 x 2 k 2 ;   k 1 , k 2 , n 0
Doc 24
0.3596, -29.0000, 11.0000, 0.6744
testing/wikipedia/v3/10096.html
( x 1 + x 2 + + x m ) n = k 1 , k 2 , , k m ( n k 1 , k 2 , , k m ) x 1 k 1 x 2 k 2 x m k m .
Doc 1
0.9365, -6.0000, 29.0000, 2.8460
testing/wikipedia/v3/00133.html
Doc 5
0.5554, -3.0000, 14.0000, 0.9150
testing/wikipedia/v3/02165.html
u n = k = 0 n ( n k ) a k ( - c ) n - k b k
Doc 25
0.3468, -9.0000, 9.0000, 0.3468
testing/wikipedia/v3/12660.html
( a b ) n = j = 0 n ( n j ) a j b n - j .
Doc 26
0.3468, -9.0000, 6.0000, 0.9488
testing/wikipedia/v3/02880.html
( p + q ) 8.3 = k = 0 ( 8.3 k ) p k q 8.3 - k .
Doc 9
0.4282, -12.0000, 9.0000, 1.1715
testing/wikipedia/v3/01438.html
( x y ) n = j = 1 n - 1 ( n j ) x j y n - j
Doc 27
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 28
0.3468, -11.0000, 7.0000, 0.3468
testing/wikipedia/v3/07606.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, 2.8460
testing/wikipedia/v3/00133.html
B n ( x ) = k = 0 n ( n n - k ) b k x n - k ,
Doc 30
0.3328, -11.0000, 8.0000, 0.3328
testing/wikipedia/v3/28068.html
( s + t ) n = k = 0 n ( n k ) ( s ) k ( t ) n - k
Doc 29
0.3328, -11.0000, 8.0000, 0.3328
testing/wikipedia/v3/06981.html
( x + y ) n = k = 0 n ( n k ) ( x ) n - k ( y ) k ,
Doc 32
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 31
0.3328, -12.0000, 10.0000, 1.5331
testing/wikipedia/v3/03168.html
B n ( x + y ) = k = 0 n ( n k ) B k ( x ) y n - k
Doc 31
0.3328, -12.0000, 10.0000, 1.5331
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, 1.6032
testing/wikipedia/v3/03031.html
p n ( x + y ) = k = 0 n ( n k ) p k ( x ) y n - k .
Doc 33
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, 1.6032
testing/wikipedia/v3/03031.html
p n ( x + y ) = k = 0 n ( n k ) p k ( x ) p n - k ( y )
Doc 27
0.3468, -10.0000, 10.0000, 0.6796
testing/wikipedia/v3/05621.html
p n ( x + y ) = k = 0 n ( n k ) p k ( x ) p n - k ( y ) .
Doc 19
0.3646, -12.0000, 11.0000, 1.2353
testing/wikipedia/v3/03081.html
Doc 26
0.3468, -9.0000, 6.0000, 0.9488
testing/wikipedia/v3/02880.html
s n ( x + y ) = k = 0 n ( n k ) p k ( x ) s n - k ( y ) .
Doc 19
0.3646, -12.0000, 11.0000, 1.2353
testing/wikipedia/v3/03081.html
T n ( λ + μ ) = k = 0 n ( n k ) T k ( λ ) T n - k ( μ ) .
Doc 13
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 34
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, 1.0433
testing/wikipedia/v3/02848.html
p n ( x ) = k = 0 n ( n k ) c k x n - k ;
Doc 33
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 35
0.3273, -28.0000, 10.0000, 0.3273
testing/wikipedia/v3/04108.html
E m ( x ) = k = 0 m ( m k ) E k 2 k ( x - 1 2 ) m - k .
Doc 31
0.3328, -12.0000, 10.0000, 1.5331
testing/wikipedia/v3/03168.html
( 1 + x ) n = k = 0 n ( n k ) x k
Doc 36
0.3148, -5.0000, 10.0000, 0.3148
testing/wikipedia/v3/02183.html
( 1 + x ) n = k = 0 n ( n k ) x k .
Doc 1
0.9365, -6.0000, 29.0000, 2.8460
testing/wikipedia/v3/00133.html
( 1 + x + 1 / x ) n = k = - n n ( n k ) 2 x k
Doc 37
0.3148, -11.0000, 10.0000, 0.6096
testing/wikipedia/v3/30027.html
( x 1 ) n = k 1 = n ( n k 1 ) x 1 k 1 ;  k 1 , n 0
Doc 24
0.3596, -29.0000, 11.0000, 0.6744
testing/wikipedia/v3/10096.html
μ n ( t + s ) = k = 0 n ( n k ) μ k ( t ) μ n - k ( s ) .
Doc 38
0.3148, -15.0000, 8.0000, 0.3148
testing/wikipedia/v3/07298.html
Var ( X ) = k = 0 n ( n k ) p k ( 1 - p ) n - k ( k - n p ) 2 = n p ( 1 - p ) ,
Doc 39
0.3148, -24.0000, 9.0000, 0.3148
testing/wikipedia/v3/01076.html
d n f = k = 0 n ( n k ) n f x k y n - k ( d x ) k ( d y ) n - k ,
Doc 40
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 41
0.3010, -11.0000, 7.0000, 0.3010
testing/wikipedia/v3/03879.html
Pr ( n  coin tosses yield heads at most  k  times ) = i = 0 k ( n i ) p i ( 1 - p ) n - i .
Doc 42
0.2948, -14.0000, 7.0000, 0.2948
testing/wikipedia/v3/07662.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 37
0.3148, -11.0000, 10.0000, 0.6096
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 43
0.2948, -33.0000, 6.0000, 0.2948
testing/wikipedia/v3/16716.html
( 1 + X ) n = k 0 ( n k ) X k ,
Doc 44
0.2827, -6.0000, 8.0000, 0.2827
testing/wikipedia/v3/00150.html
( 1 + x ) n = r = 0 ( n r ) x r .
Doc 45
0.2827, -7.0000, 8.0000, 0.2827
testing/wikipedia/v3/24901.html
( 1 + x ) n = k = 0 ( n k ) x k | x | < 1
Doc 3
0.5554, -2.0000, 17.0000, 1.0885
testing/wikipedia/v3/01569.html
B n ( x ) = k = 0 n ( n k ) b n - k x k ,
Doc 31
0.3328, -12.0000, 10.0000, 1.5331
testing/wikipedia/v3/03168.html
W ( C ; x , y ) = w = 0 n A w x w y n - w .
Doc 46
0.2827, -12.0000, 8.0000, 0.5332
testing/wikipedia/v3/09485.html
L n ( x ) = k = 0 n ( n k ) ( - 1 ) k k ! x k .
Doc 47
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 48
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 49
0.2827, -21.0000, 8.0000, 0.2827
testing/wikipedia/v3/03398.html
P ( k ) = ( n k ) p k q n - k
Doc 50
0.2692, -4.0000, 6.0000, 0.2692
testing/wikipedia/v3/02208.html
E ( X n ) = k = 1 n { n k } λ k .
Doc 51
0.2692, -7.0000, 6.0000, 0.2692
testing/wikipedia/v3/10244.html
c n = k = 0 n ( n k ) a k b n - k
Doc 26
0.3468, -9.0000, 6.0000, 0.9488
testing/wikipedia/v3/02880.html
a ( n ) = i = 1 n ( n i ) a ( n - i ) .
Doc 52
0.2692, -9.0000, 6.0000, 0.2692
testing/wikipedia/v3/15026.html
( i = 1 n x i ) k = | α | = k ( k α ) x α
Doc 12
0.3964, -7.0000, 9.0000, 0.6656
testing/wikipedia/v3/05791.html
n , k ( n k ) x k y n = 1 1 - y - x y .
Doc 2
0.5554, -2.0000, 17.0000, 1.6288
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, 1.6288
testing/wikipedia/v3/00131.html
Δ n x m = k = 0 n ( - 1 ) n - k ( n k ) ( x + k ) m
Doc 31
0.3328, -12.0000, 10.0000, 1.5331
testing/wikipedia/v3/03168.html
n , k ( n k ) x k y n = 1 1 - ( 1 + x ) y = 1 1 - y - x y .
Doc 53
0.2692, -22.0000, 7.0000, 0.2692
testing/wikipedia/v3/02490.html
k = 0 n ( n k ) k n - k
Doc 54
0.2505, -3.0000, 8.0000, 0.2505
testing/wikipedia/v3/00526.html
( x + 1 ) n = i = 0 n a i x i .
Doc 3
0.5554, -2.0000, 17.0000, 1.0885
testing/wikipedia/v3/01569.html
b n = k = 0 n ( n k ) 2 ( n + k k ) 2 .
Doc 55
0.2505, -10.0000, 6.0000, 0.2505
testing/wikipedia/v3/09155.html
A ( C ; x , y ) = i = 0 n A i x i y n - i
Doc 46
0.2827, -12.0000, 8.0000, 0.5332
testing/wikipedia/v3/09485.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, 1.0433
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 56
0.2390, -25.0000, 6.0000, 0.4247
testing/wikipedia/v3/02190.html
D ( n ) = k = 0 n ( n k ) ( n + k k ) .
Doc 57
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 58
0.2373, -14.0000, 7.0000, 0.2373
testing/wikipedia/v3/03051.html
e = i = 1 n x i y i = i = 1 n y i x i
Doc 59
0.2182, -15.0000, 7.0000, 0.2182
testing/wikipedia/v3/17881.html
n = 0 N k = 0 K ( K k ) ( N - K n - k ) x k y n - k .
Doc 21
0.3596, -5.0000, 10.0000, 0.5778
testing/wikipedia/v3/03830.html
n , k 1 ( n + k ) ! ( n + k k ) x k y n = e x + y .
Doc 2
0.5554, -2.0000, 17.0000, 1.6288
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 19
0.3646, -12.0000, 11.0000, 1.2353
testing/wikipedia/v3/03081.html
Doc 33
0.3328, -13.0000, 10.0000, 0.8651
testing/wikipedia/v3/09031.html
( 1 + x ) k = 0 n ( α k ) x k = k = 0 n ( α + 1 k ) x k + ( α n ) x n + 1 ,
Doc 60
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 56
0.2390, -25.0000, 6.0000, 0.4247
testing/wikipedia/v3/02190.html
( n - 1 k ) - ( n - 1 k - 1 ) = n - 2 k n ( n k ) .
Doc 2
0.5554, -2.0000, 17.0000, 1.6288
testing/wikipedia/v3/00131.html
x y ¯ = 1 n i = 1 n x i y i .
Doc 61
0.1623, -10.0000, 6.0000, 0.1623
testing/wikipedia/v3/02757.html
Doc 62
0.1623, -10.0000, 6.0000, 0.1623
testing/wikipedia/v3/10770.html
( n k + 1 ) = n - k k + 1 ( n k )
Doc 2
0.5554, -2.0000, 17.0000, 1.6288
testing/wikipedia/v3/00131.html
x i [ 𝐟 A ] = k = 1 n a ~ k i x k [ 𝐟 ] .
Doc 63
0.1529, -13.0000, 4.0000, 0.1529
testing/wikipedia/v3/02917.html
y ( n ) = p H p x ( n ) p G p x ( n )
Doc 64
0.1342, -12.0000, 6.0000, 0.1342
testing/wikipedia/v3/14368.html