A257050 Array a(m,n) (m>0, n>=0) of quotient of de Bruijn alternating sums of m-th powers of binomial coefficients, listed by ascending antidiagonals.

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 15, 1, 0, 1, 15, 131, 84, 1, 0, 1, 31, 955, 3067, 495, 1, 0, 1, 63, 6411, 84840, 79459, 3003, 1, 0, 1, 127, 41195, 2065603, 8765595, 2181257, 18564, 1, 0, 1, 255, 258091, 46942056, 813289963, 987430015, 62165039, 116280, 1, 0

Offset

1,8

Links

Table of n, a(n) for n=1..55.

Neil J. Calkin, Factors of sums of powers of binomial coefficients

Victor J. W. Guo, Frédéric Jouhet and Jiang Zeng, Factors of alternating sums of products of binomial and q-binomial coefficients, arXiv: math.NT/0511635 (2005-2007)

Eric Weisstein's MathWorld, Binomial Sums

Formula

a(m, n) = (Sum_{k=-n..n} (-1)^k*binomial(2*n, n+k)^m)/binomial(2*n, n).

Example

With S(s,n) = de Bruijn sum, array begins:
1,  0,   0,     0,       0,         0,            0, ...
1,  1,   1,     1,       1,         1,            1, ...
1,  3,  15,    84,     495,      3003,        18564, ... = A005809 = S(3,n)/S(2,n)
1,  7, 131,  3067,   79459,   2181257,     62165039, ... = A099601 = S(4,n)/S(2,n)
1, 15, 955, 84840, 8765595, 987430015, 117643216600, ... = S(5,n)/S(2,n)
...
Second column is A000225 (Mersenne numbers).

Mathematica

a[m_, n_] := Sum[(-1)^k*Binomial[2*n, n+k]^m, {k, -n, n}]/Binomial[2*n, n]; Table[a[m-n, n], {m, 1, 10}, {n, 0, m-1}] // Flatten

Crossrefs

Cf. A005809, A006480, A050983, A050984, A099601, A227357.

Sequence in context: A048993 A264431 A357941 * A274494 A274490 A193357

Adjacent sequences: A257047 A257048 A257049 * A257051 A257052 A257053

Keyword

nonn,tabl

Author

Jean-François Alcover, Apr 15 2015

Status

approved