Returned 98 matches (100 formulae, 64 docs)
    Lookup 105.399 ms, Re-ranking 4754.321 ms
    Found 355462 tuple postings, 130643 formulae, 18813 documents
[ formulas ] [ documents ] [ documents-by-formula ]

Doc 1
0.9365
-6.0000
29.0000
3.0832
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 + 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 .
( 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 .
cos ( n x ) = k  even ( - 1 ) k / 2 ( n k ) cos n - k x sin k x

Doc 2
0.5554
-2.0000
17.0000
1.4760
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 1 ( n + k ) ! ( n + k k ) x k y n = e x + y .
( n - 1 k ) - ( n - 1 k - 1 ) = n - 2 k n ( n k ) .

Doc 3
0.5554
-2.0000
17.0000
1.0885
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
( x + 1 ) n = i = 0 n a i x i .
Doc 4
0.5554
-2.0000
13.0000
1.6032
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 ,
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 .

Doc 5
0.5554
-3.0000
14.0000
0.9150
testing/wikipedia/v3/02165.html
( x + y ) n = i = 0 n ( n i ) x n - i y i ,
( 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 6
0.4600
-8.0000
14.0000
1.0433
testing/wikipedia/v3/02848.html
𝐻𝑒 n ( x + y ) = k = 0 n ( n k ) x n - k 𝐻𝑒 k ( y )
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
0.4427
-39.0000
12.0000
0.7118
testing/wikipedia/v3/10244.html
r = 1 k 1 1 - r x = n = 0 { n k } x n - k ,   or  n = 0 { n k } x n + 1 = 1 ( k + 1 ) ! ( 1 x k + 1 ) .
E ( X n ) = k = 1 n { n k } λ k .

Doc 8
0.4364
-13.0000
13.0000
1.5938
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 ,
Δ n = ( E - I ) n = j = 0 n ( n j ) ( - 1 ) n - j E j ,   n 0 .
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 9
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 10
0.4282
-7.0000
5.0000
0.4282
testing/wikipedia/v3/08319.html
( a + b ) p = n = 0 p ( p n ) a n b p - n .

Doc 11
0.4282
-12.0000
9.0000
1.1715
testing/wikipedia/v3/01438.html
1 = 1 n = ( p + q ) n = k = 0 n ( n k ) p k q n - k .
( p + q ) n = k = 0 ( n k ) p k q n - k ,
( p + q ) 8.3 = k = 0 ( 8.3 k ) p k q 8.3 - k .
Doc 12
0.4282
-23.0000
8.0000
0.4282
testing/wikipedia/v3/03433.html
( a + b ) n + m - 1 = i = 0 n + m - 1 ( n + m - 1 i ) a i b n + m - 1 - i .

Doc 13
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 14
0.3964
-7.0000
9.0000
0.6656
testing/wikipedia/v3/05791.html
( x + y ) α = ν α ( α ν ) x ν y α - ν .
( i = 1 n x i ) k = | α | = k ( k α ) x α

Doc 15
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 16
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 17
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 18
0.3788
-18.0000
8.0000
0.3788
testing/wikipedia/v3/15089.html
( a X + b Y ) n = i = 0 n ( n i ) a i b n - i X i Y n - i

Doc 19
0.3714
-24.0000
6.0000
0.3714
testing/wikipedia/v3/13655.html
( 1 - z ) w = n 0 ( w n ) ( - 1 ) n z n = n 0 ( - 1 ) n n ! s n ( w ) z n .

Doc 20
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 21
0.3646
-12.0000
11.0000
1.0303
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 ) .
Doc 22
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 23
0.3596
-5.0000
10.0000
0.5778
testing/wikipedia/v3/03830.html
( 1 + x ) n = k = 0 n ( n k ) x k .
n = 0 N k = 0 K ( K k ) ( N - K n - k ) x k y n - k .

Doc 24
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 25
0.3596
-13.0000
8.0000
0.3596
testing/wikipedia/v3/06439.html
S = i = n 2 + 1 n ( n i ) p i ( 1 - p ) n - i .

Doc 26
0.3596
-29.0000
11.0000
0.6744
testing/wikipedia/v3/10096.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
( x 1 ) n = k 1 = n ( n k 1 ) x 1 k 1 ;  k 1 , n 0

Doc 27
0.3468
-9.0000
9.0000
0.3468
testing/wikipedia/v3/12660.html
u n = k = 0 n ( n k ) a k ( - c ) n - k b k

Doc 28
0.3468
-9.0000
6.0000
0.6796
testing/wikipedia/v3/02880.html
( a b ) n = j = 0 n ( n j ) a j b n - j .
p n ( x + y ) = k = 0 n ( n k ) p k ( x ) p n - k ( y ) .

Doc 29
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 30
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 31
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 32
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 33
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
E n ( x + y ) = k = 0 n ( n k ) E k ( x ) y n - k
E m ( x ) = k = 0 m ( m k ) E k 2 k ( x - 1 2 ) m - 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

Doc 34
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 35
0.3328
-13.0000
10.0000
0.6601
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 ;

Doc 36
0.3328
-14.0000
7.0000
0.3328
testing/wikipedia/v3/03187.html
( a + b ) ( n ) = j = 0 n ( n j ) ( a ) ( n - j ) ( b ) ( j )

Doc 37
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 38
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 39
0.3148
-5.0000
10.0000
0.3148
testing/wikipedia/v3/02183.html
( 1 + x ) n = k = 0 n ( n k ) x k

Doc 40
0.3148
-11.0000
10.0000
0.6096
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

Doc 41
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 42
0.3148
-24.0000
9.0000
0.3148
testing/wikipedia/v3/01076.html
Var ( X ) = k = 0 n ( n k ) p k ( 1 - p ) n - k ( k - n p ) 2 = n p ( 1 - p ) ,

Doc 43
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 44
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 45
0.2948
-14.0000
7.0000
0.2948
testing/wikipedia/v3/07662.html
Pr ( n  coin tosses yield heads at most  k  times ) = i = 0 k ( n i ) p i ( 1 - p ) n - i .

Doc 46
0.2827
-6.0000
8.0000
0.2827
testing/wikipedia/v3/00150.html
( 1 + X ) n = k 0 ( n k ) X k ,

Doc 47
0.2827
-7.0000
8.0000
0.2827
testing/wikipedia/v3/24901.html
( 1 + x ) n = r = 0 ( n r ) x r .

Doc 48
0.2827
-12.0000
8.0000
0.5332
testing/wikipedia/v3/09485.html
W ( C ; x , y ) = w = 0 n A w x w y n - w .
A ( C ; x , y ) = i = 0 n A i x i y n - i

Doc 49
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 50
0.2827
-15.0000
7.0000
0.2827
testing/wikipedia/v3/22136.html
𝒫 ( t ) = k = 0 n a n - k , k t n - k ,  𝒫 ( ω ) = 0.

Doc 51
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 52
0.2692
-9.0000
6.0000
0.2692
testing/wikipedia/v3/15026.html
a ( n ) = i = 1 n ( n i ) a ( n - i ) .

Doc 53
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 54
0.2505
-3.0000
8.0000
0.2505
testing/wikipedia/v3/00526.html
k = 0 n ( n k ) k n - k

Doc 55
0.2505
-10.0000
6.0000
0.2505
testing/wikipedia/v3/09155.html
b n = k = 0 n ( n k ) 2 ( n + k k ) 2 .

Doc 56
0.2373
-10.0000
7.0000
0.2373
testing/wikipedia/v3/21377.html
D ( n ) = k = 0 n ( n k ) ( n + k k ) .

Doc 57
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 58
0.2182
-15.0000
7.0000
0.2182
testing/wikipedia/v3/17881.html
e = i = 1 n x i y i = i = 1 n y i x i

Doc 59
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 60
0.1961
-29.0000
7.0000
0.1961
testing/wikipedia/v3/09974.html
f ( x ) g ( y ) - f ( y ) g ( x ) x - y = i , j = 1 n b i j x i - 1 y j - 1 .

Doc 61
0.1623
-10.0000
6.0000
0.1623
testing/wikipedia/v3/02757.html
x y ¯ = 1 n i = 1 n x i y i .

Doc 62
0.1623
-10.0000
6.0000
0.1623
testing/wikipedia/v3/10770.html
x y ¯ = 1 n i = 1 n x i y i .

Doc 63
0.1529
-13.0000
4.0000
0.1529
testing/wikipedia/v3/02917.html
x i [ 𝐟 A ] = k = 1 n a ~ k i x k [ 𝐟 ] .

Doc 64
0.1342
-12.0000
6.0000
0.1342
testing/wikipedia/v3/14368.html
y ( n ) = p H p x ( n ) p G p x ( n )