A296775 Expansion of 1/Sum_{k>=0} A000326(k+1)*x^k.
1, -5, 13, -27, 54, -108, 216, -432, 864, -1728, 3456, -6912, 13824, -27648, 55296, -110592, 221184, -442368, 884736, -1769472, 3538944, -7077888, 14155776, -28311552, 56623104, -113246208, 226492416, -452984832, 905969664, -1811939328, 3623878656
Offset: 0
Keywords
Links
- Robert Israel, Table of n, a(n) for n = 0..3316
- Wikipedia, Pentagonal number
- Index entries for linear recurrences with constant coefficients, signature (-2).
Programs
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Magma
[1,-5,13] cat [-27*(-2)^(n-3): n in [3..50]]; // G. C. Greubel, Jan 04 2023
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Maple
1,-5,13,seq(-27*(-2)^i,i=0..50); # Robert Israel, Dec 20 2017
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Mathematica
CoefficientList[Series[1/Sum[(k+1)*(3*k+2)*x^k/2, {k, 0, 30}], {x, 0, 30}], x] (* Vaclav Kotesovec, Dec 20 2017 *) Join[{1, -5, 13}, Table[(-1)^n * 27 * 2^(n-3), {n, 3, 30}]] (* Vaclav Kotesovec, Dec 20 2017 *) LinearRecurrence[{-2},{1,-5,13,-27},40] (* Harvey P. Dale, Mar 22 2025 *) -
PARI
N=66; my(x='x+O('x^N)); Vec(1/sum(k=0, N, (k+1)*(3*k+2)/2*x^k)) -
PARI
first(n) = Vec((1-x)^3/(1+2*x) + O(x^n)) \\ Iain Fox, Dec 20 2017
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SageMath
[1,-5,13]+[-27*(-2)^(n-3) for n in range(3,51)] # G. C. Greubel, Jan 04 2023
Formula
a(n) = -2*a(n-1) for n > 3.
For n >= 3, a(n) = (-1)^n * 27 * 2^(n-3). - Vaclav Kotesovec, Dec 20 2017
G.f.: (1-x)^3/(1+2*x). - Robert Israel, Dec 20 2017
E.g.f.: (1/8)*(- 19 + 14*x - 2*x^2 + 27*exp(-2*x) ). - Alejandro J. Becerra Jr., Feb 16 2021