A383944 - OEIS

%I #18 Aug 28 2025 00:04:51

%S 1,10,87,708,5565,42798,324275,2430536,18068409,133454610,980588367,

%T 7174290060,52301288949,380120468406,2755437681195,19928252747664,

%U 143839643441265,1036380251867418,7455465737930567,53557027924956500,384241833300244269,2753539115904779070

%N Expansion of 1/sqrt((1-7*x)^3 * (1+x)).

%H Vincenzo Librandi, <a href="/A383944/b383944.txt">Table of n, a(n) for n = 0..800</a>

%F n*a(n) = (6*n+4)*a(n-1) + 7*n*a(n-2) for n > 1.

%F a(n) = (1/4)^n * Sum_{k=0..n} 7^k * (-1)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).

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

%F a(n) = Sum_{k=0..n} (-2)^k * 7^(n-k) * binomial(2*k,k) * binomial(n+1,n-k).

%t CoefficientList[Series[1/Sqrt[(1-7*x)^3*(1+x)],{x,0,33}],x] (* _Vincenzo Librandi_, Aug 27 2025 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-7*x)^3*(1+x)))

%o (Magma) R<x> := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1-7*x)^3 * (1+x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // _Vincenzo Librandi_, Aug 27 2025

%Y Cf. A322242, A383945.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Aug 19 2025