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A307911
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Coefficient of x^n in expansion of (1 - n*x - n*x^2)^n.
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3
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1, -1, 0, 27, -416, 5625, -74304, 924385, -8626176, -48361131, 7124800000, -340421390199, 13686496542720, -522760216822129, 19658830846298112, -735037915447265625, 27218267709730979840, -980444996625142158435, 32830565919734078521344, -889052809376495994642527
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OFFSET
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0,4
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COMMENTS
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Also coefficient of x^n in the expansion of 1/sqrt(1 + 2*n*x + n*(n+4)*x^2).
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LINKS
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FORMULA
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a(n) = Sum_{k=0..floor(n/2)} (-n)^(n-k) * binomial(n,k) * binomial(n-k,k) = Sum_{k=0..floor(n/2)} (-n)^(n-k) * binomial(n,2*k) * binomial(2*k,k).
For n>0, a(n) = (-n)^n * Hypergeometric2F1(1/2 - n/2, -n/2, 1, -4/n). - Vaclav Kotesovec, May 12 2021
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MATHEMATICA
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a[0] = 1; a[n_] := Sum[(-n)^(n-k) * Binomial[n, 2*k] * Binomial[2*k, k], {k, 0, Floor[n/2]}]; Array[a, 20, 0] // Flatten (* Amiram Eldar, May 12 2021 *)
Join[{1}, Table[(-n)^n*Hypergeometric2F1[1/2 - n/2, -n/2, 1, -4/n], {n, 1, 20}]] (* Vaclav Kotesovec, May 12 2021 *)
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PROG
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(PARI) {a(n) = polcoef((1-n*x-n*x^2)^n, n)}
(PARI) {a(n) = sum(k=0, n\2, (-n)^(n-k)*binomial(n, k)*binomial(n-k, k))}
(PARI) {a(n) = sum(k=0, n\2, (-n)^(n-k)*binomial(n, 2*k)*binomial(2*k, k))}
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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