A331516 - OEIS

%I #34 Sep 08 2022 08:46:25

%S 1,15,174,1850,18855,187425,1832460,17705700,169569405,1612842275,

%T 15256106778,143660483070,1347716324227,12603114069525,

%U 117536416879320,1093553079352200,10153324144411065,94098595671581175,870667876141568070,8044341506669534850

%N Expansion of 1/(1 - 10*x + 9*x^2)^(3/2).

%H Seiichi Manyama, <a href="/A331516/b331516.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = 1/2 * Sum_{k=1..n+1} 3^(n+1-k) * k * binomial(n+1,k) * binomial(n+1+k,k).

%F n * a(n) = 5 * (2*n+1) * a(n-1) - 9 * (n+1) * a(n-2) for n>1.

%F a(n) = ((n+2)/2) * Sum_{k=0..n} 4^k * binomial(n+1,k) * binomial(n+1,k+1).

%F a(n) ~ sqrt(n) * 3^(2*n + 3) / (2^(7/2) * sqrt(Pi)). - _Vaclav Kotesovec_, Jan 26 2020

%t a[n_] := 1/2 * Sum[3^( n + 1 - k) * k * Binomial[n + 1, k] * Binomial[n + 1 + k, k], {k, 1, n+1}]; Array[a, 20, 0] (* _Amiram Eldar_, Jan 20 2020 *)

%t CoefficientList[Series[1/(1-10x+9x^2)^(3/2),{x,0,20}],x] (* _Harvey P. Dale_, Nov 04 2021 *)

%o (PARI) N=20; x='x+O('x^N); Vec(1/(1-10*x+9*x^2)^(3/2))

%o (PARI) {a(n) = sum(k=1, n+1, 3^(n+1-k)*k*binomial(n+1, k)*binomial(n+1+k, k))/2}

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( 1/(1 - 10*x + 9*x^2)^(3/2))); // _Marius A. Burtea_, Jan 20 2020

%o (Magma) [1/2*&+[3^(n-k+1)*k*Binomial(n+1, k)*Binomial(n+k+1,k):k in [1..n+1]]:n in [0..20]]; // _Marius A. Burtea_, Jan 20 2020

%Y Column 5 of A331514.

%Y Cf. A084771.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 19 2020