A296817 Expansion of 1/Sum_{k>=0} (2*k+1)^2*x^k.

1, -9, 56, -328, 1912, -11144, 64952, -378568, 2206456, -12860168, 74954552, -436867144, 2546248312, -14840622728, 86497488056, -504144305608, 2938368345592, -17126065767944, 99818026262072, -581782091804488, 3390874524564856, -19763465055584648

Offset: 0

Author

Seiichi Manyama, Dec 21 2017

Keywords

• sign
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (-6,-1).

Crossrefs

Cf. A016754, A115291.

Programs

Maple

f:= gfun:-rectoproc({a(n) = -6 * a(n-1) - a(n-2), a(0)=1, a(1)=-9, a(2)=56, a(3)=-328},a(n),remember):
map(f, [$0..50]); # Robert Israel, Dec 21 2017

Mathematica

CoefficientList[Series[1/Sum[(2*k+1)^2*x^k, {k, 0, 30}], {x, 0, 30}], x] (* Vaclav Kotesovec, Dec 21 2017 *)
f[n_] := Simplify[ 4*(-1)^n*((Sqrt[2] +1)^(2n -1) - (Sqrt[2] -1)^(2n -1))]; f[0] = 1; f[1] = -9; Array[f, 22, 0] (* or *)
CoefficientList[ Series[-(x^3 -3x^2 +3x -1)/(x^2 +6x +1), {x, 0, 21}], x] (* or *)
Join[{1, -9}, LinearRecurrence[{-6, -1}, {56, -328}, 20]] (* Robert G. Wilson v, Dec 21 2017 *)

PARI

N=66; x='x+O('x^N); Vec(1/sum(k=0, N, (2*k+1)^2*x^k))

Formula

a(n) = -6 * a(n-1) - a(n-2) for n > 3.

For n > 1, a(n) = 4*(-1)^n * ((sqrt(2)+1)^(2*n-1) - (sqrt(2)-1)^(2*n-1)). - Vaclav Kotesovec, Dec 21 2017

G.f.: (1-x)^3/(1+6*x+x^2). - Robert Israel, Dec 21 2017

a(n) = 8*A002315(n-1), n>1. - R. J. Mathar, Jan 27 2020