OFFSET
1,8
LINKS
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
KEYWORD
nonn,tabl
AUTHOR
Jean-François Alcover, Apr 15 2015
STATUS
approved