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Expansion of 1/Sum_{k>=0} A000326(k+1)*x^k.
1

%I #27 Jan 05 2023 03:14:32

%S 1,-5,13,-27,54,-108,216,-432,864,-1728,3456,-6912,13824,-27648,55296,

%T -110592,221184,-442368,884736,-1769472,3538944,-7077888,14155776,

%U -28311552,56623104,-113246208,226492416,-452984832,905969664,-1811939328,3623878656

%N Expansion of 1/Sum_{k>=0} A000326(k+1)*x^k.

%H Robert Israel, <a href="/A296775/b296775.txt">Table of n, a(n) for n = 0..3316</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pentagonal_number">Pentagonal number</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (-2).

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

%F For n >= 3, a(n) = (-1)^n * 27 * 2^(n-3). - _Vaclav Kotesovec_, Dec 20 2017

%F G.f.: (1-x)^3/(1+2*x). - _Robert Israel_, Dec 20 2017

%F E.g.f.: (1/8)*(- 19 + 14*x - 2*x^2 + 27*exp(-2*x) ). - _Alejandro J. Becerra Jr._, Feb 16 2021

%p 1,-5,13,seq(-27*(-2)^i,i=0..50); # _Robert Israel_, Dec 20 2017

%t CoefficientList[Series[1/Sum[(k+1)*(3*k+2)*x^k/2, {k, 0, 30}], {x, 0, 30}], x] (* _Vaclav Kotesovec_, Dec 20 2017 *)

%t Join[{1, -5, 13}, Table[(-1)^n * 27 * 2^(n-3), {n, 3, 30}]] (* _Vaclav Kotesovec_, Dec 20 2017 *)

%o (PARI) N=66; my(x='x+O('x^N)); Vec(1/sum(k=0, N, (k+1)*(3*k+2)/2*x^k))

%o (PARI) first(n) = Vec((1-x)^3/(1+2*x) + O(x^n)) \\ _Iain Fox_, Dec 20 2017

%o (Magma) [1,-5,13] cat [-27*(-2)^(n-3): n in [3..50]]; // _G. C. Greubel_, Jan 04 2023

%o (SageMath) [1,-5,13]+[-27*(-2)^(n-3) for n in range(3,51)] # _G. C. Greubel_, Jan 04 2023

%Y Cf. A000326, A294372.

%K sign

%O 0,2

%A _Seiichi Manyama_, Dec 20 2017