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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. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,8
LINKS
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
Sequence in context: A048993 A264431 A357941 * A274494 A274490 A193357
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved

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Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)