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 ]

0.9365
-6.0000
29.0000
( x + y ) n = k = 0 n ( n k ) x n - k y k = k = 0 n ( n k ) x k y n - k .

0.5554
-2.0000
17.0000
( x + y ) n = k = 0 n ( n k ) x n - k y k

0.5554
-2.0000
13.0000
( y + x ) n = k = 0 n ( n k ) y n - k x k

0.5554
-3.0000
14.0000
( x + y ) n = i = 0 n ( n i ) x n - i y i ,

0.4918
-13.0000
15.0000
( x + y ) n + 1 = k = 0 n + 1 ( n + 1 k ) x n + 1 - k y k ,

0.4600
-8.0000
14.0000
𝐻𝑒 n ( x + y ) = k = 0 n ( n k ) x n - k 𝐻𝑒 k ( y )

0.4364
-13.0000
13.0000
( Δ n c ) 0 = k = 0 n ( n k ) ( - 1 ) n - k z k = ( z - 1 ) n .

0.4282
-7.0000
5.0000
( a + b ) p = n = 0 p ( p n ) a n b p - n .

0.4282
-12.0000
9.0000
1 = 1 n = ( p + q ) n = k = 0 n ( n k ) p k q n - k .

0.4108
-6.0000
5.0000
B k ( j ) = n = 0 k ( k n ) B n j k - n .

0.4108
-37.0000
11.0000
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 !

0.4039
-11.0000
12.0000
Δ n = ( E - I ) n = k = 0 n ( n k ) ( - 1 ) n - k E k ,

0.3964
-7.0000
9.0000
( x + y ) α = ν α ( α ν ) x ν y α - ν .

0.3964
-8.0000
8.0000
( p + q ) n = k = 0 ( n k ) p k q n - k ,

0.3919
-26.0000
12.0000
T 0 ( x ) = 1 ,   T n ( x ) = k = 1 n S ( n , k ) x k = k = 1 n { n k } x k ,  n > 0 ,

0.3822
-11.0000
14.0000
B n ( x ) = k = 0 n ( n k ) b n - k x k = ( b + x ) n ,

0.3822
-16.0000
9.0000
Δ n = ( E - I ) n = j = 0 n ( n j ) ( - 1 ) n - j E j ,   n 0 .

0.3822
-35.0000
14.0000
( 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 ,

0.3788
-6.0000
7.0000
B n ( x ) = j = 0 n ( n j ) B j x n - j

0.3788
-18.0000
8.0000
( a X + b Y ) n = i = 0 n ( n i ) a i b n - i X i Y n - i

0.3714
-24.0000
6.0000
( 1 - z ) w = n 0 ( w n ) ( - 1 ) n z n = n 0 ( - 1 ) n n ! s n ( w ) z n .

0.3714
-33.0000
9.0000
j = k n ( n j ) ( - 1 ) n - j = - j = 0 k - 1 ( n j ) ( - 1 ) n - j = ( - 1 ) n - k ( n - 1 k - 1 ) .

0.3646
-8.0000
9.0000
( x 1 + + x m ) n = | α | = n ( n α ) x α

0.3646
-12.0000
11.0000
s n ( x + y ) = k = 0 n ( n k ) x k s n - k ( y ) .

0.3646
-19.0000
9.0000
1 e k = x k n ( k - x ) ! = k = 0 n ( n k ) B k x n - k

0.3596
-5.0000
10.0000
( 1 + x ) n = k = 0 n ( n k ) x k .

0.3596
-13.0000
11.0000
Δ n [ f ] ( x ) = k = 0 n ( n k ) ( - 1 ) n - k f ( x + k )

0.3596
-13.0000
8.0000
S = i = n 2 + 1 n ( n i ) p i ( 1 - p ) n - i .

0.3596
-29.0000
11.0000
( 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

0.3596
-37.0000
11.0000
( 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 .

0.3468
-9.0000
9.0000
u n = k = 0 n ( n k ) a k ( - c ) n - k b k

0.3468
-9.0000
6.0000
( a b ) n = j = 0 n ( n j ) a j b n - j .

0.3468
-9.0000
6.0000
( p + q ) 8.3 = k = 0 ( 8.3 k ) p k q 8.3 - k .

0.3468
-10.0000
10.0000
( x y ) n = j = 1 n - 1 ( n j ) x j y n - j

0.3468
-11.0000
7.0000
( f g ) ( n ) = k = 0 n ( n k ) f ( k ) g ( n - k )

0.3468
-55.0000
10.0000
( 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 .

0.3328
-11.0000
8.0000
B n ( x ) = k = 0 n ( n n - k ) b k x n - k ,

0.3328
-11.0000
8.0000
( s + t ) n = k = 0 n ( n k ) ( s ) k ( t ) n - k

0.3328
-12.0000
10.0000
( x + y ) n = k = 0 n ( n k ) ( x ) n - k ( y ) k ,

0.3328
-12.0000
10.0000
E n ( x + y ) = k = 0 n ( n k ) E k ( x ) y n - k
B n ( x + y ) = k = 0 n ( n k ) B k ( x ) y n - k

0.3328
-12.0000
8.0000
B n ( y + x ) = k = 0 n ( n k ) B n - k ( y ) x k

0.3328
-13.0000
10.0000
p n ( x + y ) = k = 0 n ( n k ) p k ( x ) y n - k .

0.3328
-13.0000
8.0000
B n ( y + x ) = k = 0 n ( n k ) B n - k ( y ) x k .

0.3328
-14.0000
10.0000
p n ( x + y ) = k = 0 n ( n k ) p k ( x ) p n - k ( y )

0.3328
-15.0000
10.0000
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 ) .

0.3328
-15.0000
8.0000
T n ( λ + μ ) = k = 0 n ( n k ) T k ( λ ) T n - k ( μ ) .

0.3328
-17.0000
9.0000
( x ) ( n ) = x ( x + 1 ) ( x + n - 1 ) = k = 0 n [ n k ] x k

0.3328
-23.0000
10.0000
𝐻𝑒 n [ α + β ] ( x + y ) = k = 0 n ( n k ) 𝐻𝑒 k [ α ] ( x ) 𝐻𝑒 n - k [ β ] ( y ) .

0.3273
-8.0000
9.0000
p n ( x ) = k = 0 n ( n k ) c k x n - k ;

0.3273
-28.0000
10.0000
ν = 0 n b ν , n ( x ) = ν = 0 n ( n ν ) x ν ( 1 - x ) n - ν = ( x + ( 1 - x ) ) n = 1.

0.3156
-15.0000
7.0000
E m ( x ) = k = 0 m ( m k ) E k 2 k ( x - 1 2 ) m - k .

0.3148
-5.0000
10.0000
( 1 + x ) n = k = 0 n ( n k ) x k

0.3148
-6.0000
10.0000
( 1 + x ) n = k = 0 n ( n k ) x k .

0.3148
-11.0000
10.0000
( 1 + x + 1 / x ) n = k = - n n ( n k ) 2 x k

0.3148
-15.0000
9.0000
( x 1 ) n = k 1 = n ( n k 1 ) x 1 k 1 ;  k 1 , n 0

0.3148
-15.0000
8.0000
μ n ( t + s ) = k = 0 n ( n k ) μ k ( t ) μ n - k ( s ) .

0.3148
-24.0000
9.0000
Var ( X ) = k = 0 n ( n k ) p k ( 1 - p ) n - k ( k - n p ) 2 = n p ( 1 - p ) ,

0.3148
-24.0000
7.0000
d n f = k = 0 n ( n k ) n f x k y n - k ( d x ) k ( d y ) n - k ,

0.3010
-11.0000
7.0000
Δ ( X n ) = k = 0 n ( n k ) X k X n - k ,

0.2948
-14.0000
7.0000
Pr ( n  coin tosses yield heads at most  k  times ) = i = 0 k ( n i ) p i ( 1 - p ) n - i .

0.2948
-27.0000
8.0000
( 1 + x + x 2 ) n = j = 0 2 n ( n j - n ) 2 x j = k = - n n ( n k ) 2 x n + k

0.2948
-33.0000
6.0000
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 )

0.2827
-6.0000
8.0000
( 1 + X ) n = k 0 ( n k ) X k ,

0.2827
-7.0000
8.0000
( 1 + x ) n = r = 0 ( n r ) x r .

0.2827
-10.0000
9.0000
( 1 + x ) n = k = 0 ( n k ) x k | x | < 1

0.2827
-10.0000
9.0000
B n ( x ) = k = 0 n ( n k ) b n - k x k ,

0.2827
-12.0000
8.0000
W ( C ; x , y ) = w = 0 n A w x w y n - w .

0.2827
-13.0000
9.0000
L n ( x ) = k = 0 n ( n k ) ( - 1 ) k k ! x k .

0.2827
-18.0000
6.0000
( x ) m ( x ) n = k = 0 m ( m k ) ( n k ) k ! ( x ) m + n - k .

0.2827
-21.0000
8.0000
( u v ) ( n ) ( x ) = k = 0 n ( n k ) u ( n - k ) ( x ) v ( k ) ( x ) .

0.2692
-4.0000
6.0000
P ( k ) = ( n k ) p k q n - k

0.2692
-7.0000
6.0000
E ( X n ) = k = 1 n { n k } λ k .

0.2692
-8.0000
6.0000
c n = k = 0 n ( n k ) a k b n - k

0.2692
-9.0000
6.0000
a ( n ) = i = 1 n ( n i ) a ( n - i ) .

0.2692
-11.0000
5.0000
( i = 1 n x i ) k = | α | = k ( k α ) x α

0.2692
-12.0000
7.0000
n , k ( n k ) x k y n = 1 1 - y - x y .

0.2692
-13.0000
7.0000
n , k ( n + k k ) x k y n = 1 1 - x - y .

0.2692
-15.0000
4.0000
Δ n x m = k = 0 n ( - 1 ) n - k ( n k ) ( x + k ) m

0.2692
-22.0000
7.0000
n , k ( n k ) x k y n = 1 1 - ( 1 + x ) y = 1 1 - y - x y .

0.2505
-3.0000
8.0000
k = 0 n ( n k ) k n - k

0.2505
-7.0000
8.0000
( x + 1 ) n = i = 0 n a i x i .

0.2505
-10.0000
6.0000
b n = k = 0 n ( n k ) 2 ( n + k k ) 2 .

0.2505
-12.0000
7.0000
A ( C ; x , y ) = i = 0 n A i x i y n - i

0.2505
-30.0000
5.0000
k = 0 n ( n k ) 𝐻𝑒 k [ α ] ( x ) 𝐻𝑒 n - k [ - α ] ( y ) = 𝐻𝑒 n [ 0 ] ( x + y ) = ( x + y ) n .

0.2390
-25.0000
6.0000
P n ( x ) = k = 0 n ( - 1 ) k ( n k ) 2 ( 1 + x 2 ) n - k ( 1 - x 2 ) k .

0.2373
-10.0000
7.0000
D ( n ) = k = 0 n ( n k ) ( n + k k ) .

0.2373
-14.0000
7.0000
H n ( x ) = k = 0 n ( n + k n - k ) ( - x ) k .

0.2182
-15.0000
7.0000
e = i = 1 n x i y i = i = 1 n y i x i

0.2182
-22.0000
3.0000
n = 0 N k = 0 K ( K k ) ( N - K n - k ) x k y n - k .

0.2054
-19.0000
6.0000
n , k 1 ( n + k ) ! ( n + k k ) x k y n = e x + y .

0.2050
-25.0000
6.0000
p n ( x ) = k = 0 n a n , k x k and q n ( x ) = k = 0 n b n , k x k .

0.1961
-28.0000
7.0000
( 1 + x ) k = 0 n ( α k ) x k = k = 0 n ( α + 1 k ) x k + ( α n ) x n + 1 ,

0.1857
-21.0000
6.0000
P n ~ ( x ) = ( - 1 ) n k = 0 n ( n k ) ( n + k k ) ( - x ) k .

0.1769
-15.0000
8.0000
( n - 1 k ) - ( n - 1 k - 1 ) = n - 2 k n ( n k ) .

0.1623
-10.0000
6.0000
x y ¯ = 1 n i = 1 n x i y i .

0.1529
-10.0000
6.0000
( n k + 1 ) = n - k k + 1 ( n k )

0.1529
-13.0000
4.0000
x i [ 𝐟 A ] = k = 1 n a ~ k i x k [ 𝐟 ] .

0.1342
-12.0000
6.0000
y ( n ) = p H p x ( n ) p G p x ( n )