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A331551
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Expansion of (1 + 3*x)/(1 + 2*x + 9*x^2)^(3/2).
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2
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1, 0, -15, 32, 105, -576, 105, 5760, -13167, -30400, 194337, -104160, -1685255, 4497024, 7011225, -57705984, 51497505, 445080960, -1402731183, -1348950240, 16032154761, -20039110080, -110074987575, 412984420992, 190753103025
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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) = Sum_{k=0..n} (-3)^(n-k) * (n+k+1) * binomial(n,k) * binomial(n+k,k).
a(n) = Sum_{k=0..n} (-2)^k * (k+1) * binomial(n+1,k+1)^2.
a(n) = (n + 1)^2*hypergeom([-n, -n], [2], -2). - Peter Luschny, Jan 20 2020
n * (2*n-1) * a(n) = 4 * (-n^2 + 1) * a(n-1) - 9 * n * (2*n+1) * a(n-2) for n>1. - Seiichi Manyama, Jan 25 2020
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MAPLE
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a := n -> (n + 1)^2*hypergeom([-n, -n], [2], -2):
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MATHEMATICA
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a[n_] := Sum[(-2)^k * (k + 1) * Binomial[n + 1, k + 1]^2, {k, 0, n}]; Array[a, 25, 0] (* Amiram Eldar, Jan 20 2020 *)
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PROG
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(PARI) N=20; x='x+O('x^N); Vec((1+3*x)/(1+2*x+9*x^2)^(3/2))
(PARI) {a(n) = sum(k=0, n, (-3)^(n-k)*(n+k+1)*binomial(n, k)*binomial(n+k, k))}
(PARI) {a(n) = sum(k=0, n, (-2)^k*(k+1)*binomial(n+1, k+1)^2)}
(Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1 + 3*x)/(1 + 2*x + 9*x^2)^(3/2))); // Marius A. Burtea, Jan 20 2020
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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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