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