cp's OEIS Frontend

A331551 Expansion of (1 + 3*x)/(1 + 2*x + 9*x^2)^(3/2).

%I A331551 #29 Sep 08 2022 08:46:25
%S A331551 1,0,-15,32,105,-576,105,5760,-13167,-30400,194337,-104160,-1685255,
%T A331551 4497024,7011225,-57705984,51497505,445080960,-1402731183,-1348950240,
%U A331551 16032154761,-20039110080,-110074987575,412984420992,190753103025
%N A331551 Expansion of (1 + 3*x)/(1 + 2*x + 9*x^2)^(3/2).
%H A331551 Seiichi Manyama, <a href="/A331551/b331551.txt">Table of n, a(n) for n = 0..1000</a>
%F A331551 a(n) = Sum_{k=0..n} (-3)^(n-k) * (n+k+1) * binomial(n,k) * binomial(n+k,k).
%F A331551 a(n) = Sum_{k=0..n} (-2)^k * (k+1) * binomial(n+1,k+1)^2.
%F A331551 a(n) = (n + 1)^2*hypergeom([-n, -n], [2], -2). - _Peter Luschny_, Jan 20 2020
%F A331551 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
%p A331551 a := n -> (n + 1)^2*hypergeom([-n, -n], [2], -2):
%p A331551 seq(simplify(a(n)), n=0..19); # _Peter Luschny_, Jan 20 2020
%t A331551 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 *)
%o A331551 (PARI) N=20; x='x+O('x^N); Vec((1+3*x)/(1+2*x+9*x^2)^(3/2))
%o A331551 (PARI) {a(n) = sum(k=0, n, (-3)^(n-k)*(n+k+1)*binomial(n, k)*binomial(n+k, k))}
%o A331551 (PARI) {a(n) = sum(k=0, n, (-2)^k*(k+1)*binomial(n+1, k+1)^2)}
%o A331551 (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
%Y A331551 Column 0 of A331511.
%K A331551 sign
%O A331551 0,3
%A A331551 _Seiichi Manyama_, Jan 20 2020