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 ]

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 = k = 0 n ( n k ) x k y n - k .
( x + y ) n + 1 = k = 0 n + 1 ( n + 1 k ) x n + 1 - k y k ,
( x + y ) α = ν α ( α ν ) x ν y α - ν .
( 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 .
( 1 + x ) n = k = 0 n ( n k ) x k .

Doc 2
0.5554
-2.0000
17.0000
2.2011
testing/wikipedia/v3/00131.html
( x + y ) n = k = 0 n ( n k ) x n - k y k
n , k ( n k ) x k y n = 1 1 - y - x y .
n , k ( n + k k ) x k y n = 1 1 - x - y .
n = k ( n k ) y n = y k ( 1 - y ) k + 1 .
n , k 1 ( n + k ) ! ( n + k k ) x k y n = e x + y .
k ( n k ) x k = ( 1 + x ) n .
k = 0 n ( n k ) ( n n - k ) = ( 2 n n ) .
k = q n ( n k ) ( k q ) = 2 n - q ( n q )
( n h ) ( n - h k ) = ( n k ) ( n - k h )
( n k 1 , k 2 ) = ( n k 1 , n - k 1 ) = ( n k 1 ) = ( n k 2 ) .

Doc 3
0.5554
-2.0000
17.0000
0.8380
testing/wikipedia/v3/01569.html
( x + y ) n = k = 0 n ( n k ) x n - k y k
( 1 + x ) n = k = 0 ( n k ) x k | x | < 1

Doc 4
0.5554
-2.0000
13.0000
2.6088
testing/wikipedia/v3/03031.html
( y + x ) n = k = 0 n ( n k ) y n - k x k
B n ( x ) = k = 0 n ( n k ) b n - k x k = ( b + x ) n ,
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 ) .
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 ) .
B n ( y + x ) = k = 0 n ( n k ) B n - k ( y ) x k
B n ( y + x ) = k = 0 n ( n k ) B n - k ( y ) x k .
< L 1 L 2 | x n > = k = 0 n ( n k ) < L 1 | x k > < L 2 | x n - k > .

Doc 5
0.5554
-3.0000
14.0000
0.5554
testing/wikipedia/v3/02165.html
( x + y ) n = i = 0 n ( n i ) x n - i y i ,

Doc 6
0.4600
-8.0000
14.0000
1.3902
testing/wikipedia/v3/02848.html
𝐻𝑒 n ( x + y ) = k = 0 n ( n k ) x n - k 𝐻𝑒 k ( y )
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 ) .
𝐻𝑒 n [ α + β ] ( x + y ) = k = 0 n ( n k ) 𝐻𝑒 k [ α ] ( x ) 𝐻𝑒 n - k [ β ] ( y ) .
k = 0 n ( n k ) 𝐻𝑒 k [ α ] ( x ) 𝐻𝑒 n - k [ - α ] ( y ) = 𝐻𝑒 n [ 0 ] ( x + y ) = ( x + y ) n .

Doc 7
0.4364
-13.0000
13.0000
1.0908
testing/wikipedia/v3/19294.html
( Δ n c ) 0 = k = 0 n ( n k ) ( - 1 ) n - k z k = ( z - 1 ) n .
Δ n = ( E - I ) n = k = 0 n ( n k ) ( - 1 ) n - k E k ,
( N k ) = n = k m S n j = k n ( n j ) ( - 1 ) n - j .
Doc 8
0.4364
-60.0000
15.0000
0.8611
testing/wikipedia/v3/02190.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 ) ,
P n ( x ) = k = 0 n ( - 1 ) k ( n k ) 2 ( 1 + x 2 ) n - k ( 1 - x 2 ) k .
P n ~ ( x ) = ( - 1 ) n k = 0 n ( n k ) ( n + k k ) ( - x ) k .
Doc 9
0.4282
-12.0000
9.0000
0.4282
testing/wikipedia/v3/01438.html
1 = 1 n = ( p + q ) n = k = 0 n ( n k ) p k q n - k .

Doc 10
0.4108
-37.0000
11.0000
0.4108
testing/wikipedia/v3/06782.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
0.3964
-7.0000
9.0000
0.3964
testing/wikipedia/v3/05791.html
( x + y ) α = ν α ( α ν ) x ν y α - ν .

Doc 12
0.3919
-26.0000
12.0000
0.7247
testing/wikipedia/v3/04139.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 ,
T n ( λ + μ ) = k = 0 n ( n k ) T k ( λ ) T n - k ( μ ) .

Doc 13
0.3822
-35.0000
14.0000
0.3822
testing/wikipedia/v3/02327.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
0.3788
-6.0000
7.0000
0.3788
testing/wikipedia/v3/04615.html
B n ( x ) = j = 0 n ( n j ) B j x n - j

Doc 15
0.3646
-8.0000
9.0000
0.3646
testing/wikipedia/v3/05760.html
( x 1 + + x m ) n = | α | = n ( n α ) x α

Doc 16
0.3646
-12.0000
11.0000
1.2353
testing/wikipedia/v3/03081.html
s n ( x + y ) = k = 0 n ( n k ) x k s n - k ( y ) .
p n ( x + y ) = k = 0 n ( n k ) p k ( x ) p n - k ( y ) .
s n ( x + y ) = k = 0 n ( n k ) p k ( x ) s n - k ( y ) .
p n ( x ) = k = 0 n a n , k x k and q n ( x ) = k = 0 n b n , k x k .

Doc 17
0.3646
-19.0000
9.0000
0.3646
testing/wikipedia/v3/09424.html
1 e k = x k n ( k - x ) ! = k = 0 n ( n k ) B k x n - k

Doc 18
0.3596
-5.0000
10.0000
0.3596
testing/wikipedia/v3/03830.html
( 1 + x ) n = k = 0 n ( n k ) x k .

Doc 19
0.3596
-13.0000
11.0000
0.3596
testing/wikipedia/v3/12635.html
Δ n [ f ] ( x ) = k = 0 n ( n k ) ( - 1 ) n - k f ( x + k )

Doc 20
0.3468
-9.0000
9.0000
0.7163
testing/wikipedia/v3/12660.html
u n = k = 0 n ( n k ) a k ( - c ) n - k b k
𝔍 k ( a ) n = b n = i = 0 n ( - k ) n - i ( n i ) a i .
t n = k = 0 n ( - 1 ) n - k ( n k ) a k
Doc 21
0.3468
-10.0000
10.0000
0.6796
testing/wikipedia/v3/05621.html
( x y ) n = j = 1 n - 1 ( n j ) x j y n - j
p n ( x + y ) = k = 0 n ( n k ) p k ( x ) p n - k ( y )

Doc 22
0.3468
-11.0000
7.0000
0.3468
testing/wikipedia/v3/07606.html
( f g ) ( n ) = k = 0 n ( n k ) f ( k ) g ( n - k )

Doc 23
0.3328
-11.0000
8.0000
0.3328
testing/wikipedia/v3/06981.html
( s + t ) n = k = 0 n ( n k ) ( s ) k ( t ) n - k

Doc 24
0.3328
-11.0000
8.0000
0.3328
testing/wikipedia/v3/28068.html
B n ( x ) = k = 0 n ( n n - k ) b k x 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
E n ( x + y ) = k = 0 n ( n k ) E k ( x ) y n - k
B n ( x ) = k = 0 n ( n k ) b n - k x k ,
Δ n x m = k = 0 n ( - 1 ) n - k ( n k ) ( x + k ) m
x n = E n ( x ) + 1 2 k = 0 n - 1 ( n k ) E k ( x ) .

Doc 26
0.3328
-12.0000
10.0000
0.3328
testing/wikipedia/v3/01177.html
( x + y ) n = k = 0 n ( n k ) ( x ) n - k ( y ) k ,

Doc 27
0.3328
-13.0000
10.0000
0.8651
testing/wikipedia/v3/09031.html
p n ( x + y ) = k = 0 n ( n k ) p k ( x ) y n - k .
p n ( x ) = k = 0 n ( n k ) c k x n - k ;
p n ( x ) = k = 0 n a n , k x k and q n ( x ) = k = 0 n b n , k x k .
Doc 28
0.3328
-15.0000
10.0000
0.6020
testing/wikipedia/v3/02880.html
p n ( x + y ) = k = 0 n ( n k ) p k ( x ) p n - k ( y ) .
c n = k = 0 n ( n k ) a k b n - k

Doc 29
0.3328
-17.0000
9.0000
0.3328
testing/wikipedia/v3/10245.html
( x ) ( n ) = x ( x + 1 ) ( x + n - 1 ) = k = 0 n [ n k ] x k

Doc 30
0.3273
-28.0000
10.0000
0.3273
testing/wikipedia/v3/04108.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 31
0.3148
-5.0000
10.0000
0.4343
testing/wikipedia/v3/02183.html
( 1 + x ) n = k = 0 n ( n k ) x k
( x y ) n = x n y n

Doc 32
0.3148
-11.0000
10.0000
0.7830
testing/wikipedia/v3/30027.html
( 1 + x + 1 / x ) n = k = - n n ( n k ) 2 x k
( 1 + x + x 2 ) n = j = 0 2 n ( n j - n ) 2 x j = k = - n n ( n k ) 2 x n + k
( 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 33
0.3148
-15.0000
9.0000
0.3148
testing/wikipedia/v3/10096.html
( x 1 ) n = k 1 = n ( n k 1 ) x 1 k 1 ;  k 1 , n 0

Doc 34
0.3148
-15.0000
8.0000
0.3148
testing/wikipedia/v3/07298.html
μ n ( t + s ) = k = 0 n ( n k ) μ k ( t ) μ n - k ( s ) .

Doc 35
0.3148
-24.0000
7.0000
0.3148
testing/wikipedia/v3/24576.html
d n f = k = 0 n ( n k ) n f x k y n - k ( d x ) k ( d y ) n - k ,

Doc 36
0.3010
-11.0000
7.0000
0.3010
testing/wikipedia/v3/03879.html
Δ ( X n ) = k = 0 n ( n k ) X k X n - k ,

Doc 37
0.3010
-21.0000
6.0000
0.3010
testing/wikipedia/v3/01767.html
cos n x = k = 0 n ( n k ) cos k x sin n - k x cos ( 1 2 ( n - k ) π ) .

Doc 38
0.2948
-33.0000
6.0000
0.2948
testing/wikipedia/v3/16716.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 39
0.2827
-12.0000
5.0000
0.2827
testing/wikipedia/v3/15969.html
Δ n a 0 = k = 0 n ( - 1 ) k ( n k ) a n - k .

Doc 40
0.2827
-13.0000
9.0000
0.2827
testing/wikipedia/v3/07059.html
L n ( x ) = k = 0 n ( n k ) ( - 1 ) k k ! x k .

Doc 41
0.2827
-18.0000
6.0000
0.2827
testing/wikipedia/v3/03187.html
( x ) m ( x ) n = k = 0 m ( m k ) ( n k ) k ! ( x ) m + n - k .

Doc 42
0.2827
-21.0000
8.0000
0.2827
testing/wikipedia/v3/03398.html
( u v ) ( n ) ( x ) = k = 0 n ( n k ) u ( n - k ) ( x ) v ( k ) ( x ) .

Doc 43
0.2692
-6.0000
6.0000
0.2692
testing/wikipedia/v3/07188.html
k = 0 n ( n k ) A k B n - k .

Doc 44
0.2692
-22.0000
7.0000
0.2692
testing/wikipedia/v3/02490.html
n , k ( n k ) x k y n = 1 1 - ( 1 + x ) y = 1 1 - y - x y .

Doc 45
0.2505
-3.0000
8.0000
0.2505
testing/wikipedia/v3/00526.html
k = 0 n ( n k ) k n - k

Doc 46
0.2505
-10.0000
6.0000
0.4034
testing/wikipedia/v3/09155.html
b n = k = 0 n ( n k ) 2 ( n + k k ) 2 .
a n = k = 0 n c n , k ( n k ) 2 ( n + k k ) 2

Doc 47
0.2373
-10.0000
7.0000
0.2373
testing/wikipedia/v3/21377.html
D ( n ) = k = 0 n ( n k ) ( n + k k ) .

Doc 48
0.2373
-14.0000
7.0000
0.2373
testing/wikipedia/v3/03051.html
H n ( x ) = k = 0 n ( n + k n - k ) ( - x ) k .

Doc 49
0.2293
-27.0000
4.0000
0.2293
testing/wikipedia/v3/22994.html
M n ( x , β , γ ) = k = 0 n ( - 1 ) k ( n k ) ( x k ) k ! ( x - β ) n - k γ - k

Doc 50
0.2054
-17.0000
5.0000
0.2054
testing/wikipedia/v3/02101.html
( x ) n = k = 1 n ( - 1 ) n - k L ( n , k ) x ( k ) .

Doc 51
0.2050
-47.0000
7.0000
0.2050
testing/wikipedia/v3/18569.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 52
0.1961
-28.0000
7.0000
0.1961
testing/wikipedia/v3/06249.html
( 1 + x ) k = 0 n ( α k ) x k = k = 0 n ( α + 1 k ) x k + ( α n ) x n + 1 ,

Doc 53
0.1734
-13.0000
5.0000
0.1734
testing/wikipedia/v3/03752.html
k = d n ( n k ) ( k d ) = 2 n - d ( n d )

Doc 54
0.1734
-22.0000
3.0000
0.1734
testing/wikipedia/v3/02934.html
a n = k = 1 n ( - 1 ) k - 1 ( n k ) 2 k ( n - k ) a n - k .

Doc 55
0.1529
-33.0000
4.0000
0.1529
testing/wikipedia/v3/21554.html
f X ( x ; n ) = 1 2 ( n - 1 ) ! k = 0 n ( - 1 ) k ( n k ) ( x - k ) n - 1 sgn ( x - k )

Doc 56
0.1529
-34.0000
4.0000
0.1529
testing/wikipedia/v3/14766.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 57
0.1529
-35.0000
4.0000
0.1529
testing/wikipedia/v3/26465.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 58
0.1277
-25.0000
4.0000
0.1277
testing/wikipedia/v3/16562.html
C n = 2 ( n 2 ) - 1 n k = 1 n - 1 k ( n k ) 2 ( n - k 2 ) C k .

Doc 59
0.1195
-12.0000
5.0000
0.1195
testing/wikipedia/v3/00228.html
( x y ) n = x n y n [ y , x ] ( n 2 ) .