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A331795
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a(n) = Sum_{k=0..n} (-n)^k * binomial(n+1,k) * binomial(n+1,k+1).
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3
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1, 0, -3, 40, -515, 7056, -102935, 1554288, -22775319, 265497760, 586651461, -230587852560, 13426823564869, -637734419560224, 28594259589697425, -1264238490602458784, 56015489395280490385, -2499557487903373341888, 112150411888789509887053
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = [x^n] 2/(1 + 2*(n-1)*x + ((n+1)*x)^2 + (1 + (n-1)*x) * sqrt(1 + 2*(n-1)*x + ((n+1)*x)^2)).
a(n) = (n+1) * 2F1(-1 - n, -n; 2; -n), where 2F1 is the hypergeometric function. - Vaclav Kotesovec, Jan 26 2020
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MATHEMATICA
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Flatten[{1, Table[Sum[(-1)^k * n^k * Binomial[n+1, k] * Binomial[n+1, k+1], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jan 26 2020 *)
Table[(n+1) * Hypergeometric2F1[-1 - n, -n, 2, -n], {n, 0, 20}] (* Vaclav Kotesovec, Jan 26 2020 *)
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PROG
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(PARI) {a(n) = sum(k=0, n, (-n)^k*binomial(n+1, k)*binomial(n+1, k+1))}
(PARI) {a(n) = polcoef(2/(1+2*(n-1)*x+((n+1)*x)^2+(1+(n-1)*x)*sqrt(1+2*(n-1)*x+((n+1)*x)^2)), n)}
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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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