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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.

%I A257050 #11 Feb 16 2025 08:33:25
%S A257050 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,
%T A257050 1,63,6411,84840,79459,3003,1,0,1,127,41195,2065603,8765595,2181257,
%U A257050 18564,1,0,1,255,258091,46942056,813289963,987430015,62165039,116280,1,0
%N 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.
%H A257050 Neil J. Calkin, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa86/aa8612.pdf">Factors of sums of powers of binomial coefficients</a>
%H A257050 Victor J. W. Guo, Frédéric Jouhet and Jiang Zeng, <a href="http://arxiv.org/abs/math/0511635">Factors of alternating sums of products of binomial and q-binomial coefficients</a>, arXiv: math.NT/0511635 (2005-2007)
%H A257050 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/BinomialSums.html">Binomial Sums</a>
%F A257050 a(m, n) = (Sum_{k=-n..n} (-1)^k*binomial(2*n, n+k)^m)/binomial(2*n, n).
%e A257050 With S(s,n) = de Bruijn sum, array begins:
%e A257050 1,  0,   0,     0,       0,         0,            0, ...
%e A257050 1,  1,   1,     1,       1,         1,            1, ...
%e A257050 1,  3,  15,    84,     495,      3003,        18564, ... = A005809 = S(3,n)/S(2,n)
%e A257050 1,  7, 131,  3067,   79459,   2181257,     62165039, ... = A099601 = S(4,n)/S(2,n)
%e A257050 1, 15, 955, 84840, 8765595, 987430015, 117643216600, ... = S(5,n)/S(2,n)
%e A257050 ...
%e A257050 Second column is A000225 (Mersenne numbers).
%t A257050 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
%Y A257050 Cf. A005809, A006480, A050983, A050984, A099601, A227357.
%K A257050 nonn,tabl
%O A257050 1,8
%A A257050 _Jean-François Alcover_, Apr 15 2015