A308399 Expansion of 1 / Sum_{k=-oo..oo} (-x)^(k*(4*k + 1)).
1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 3, 1, 3, 3, 2, 6, 3, 4, 8, 4, 9, 9, 6, 15, 10, 12, 20, 12, 22, 23, 18, 35, 26, 30, 46, 32, 51, 54, 45, 76, 62, 71, 99, 76, 111, 117, 104, 160, 136, 154, 205, 167, 230, 244, 223, 319, 286, 319, 406, 349, 456, 484, 458, 619, 570, 632, 779, 695
Offset: 0
Keywords
Examples
For n=23 the a(23)=6 solutions are 3+3+3+3+3+3+5, 3+3+3+3+3+8, 3+3+3+3+11, 3+5+5+5+5, 5+5+5+8, and 5+5+13.
Links
- Ludovic Schwob, Table of n, a(n) for n = 0..10000
Programs
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Mathematica
nmax = 68; CoefficientList[Series[1/Sum[(-x)^(k (4 k + 1)), {k, -nmax, nmax}], {x, 0, nmax}], x] nmax = 68; CoefficientList[Series[Product[1/((1 - x^(8 k - 5)) (1 - x^(8 k - 3)) (1 - x^(8 k))), {k, 1, nmax}], {x, 0, nmax}], x] nmax = 68; CoefficientList[Series[Sum[PartitionsP[k] (-x)^k, {k, 0, nmax}]/Sum[PartitionsQ[2 k] (-x)^k, {k, 0, nmax}], {x, 0, nmax}], x]
Formula
G.f.: 1 / Sum_{k>=0} (-x)^A074378(k).
G.f.: Product_{k>=1} 1 / ((1 - x^(8*k - 5)) * (1 - x^(8*k - 3)) * (1 - x^(8*k))).
a(n) ~ sqrt(sqrt(2) - 1) * exp(sqrt(n)*Pi/2) / (2^(9/4)*n). - Vaclav Kotesovec, May 25 2019
a(n) = a(n-3) + a(n-5) - a(n-14) - a(n-18) + + - - (with a(0)=1 and a(n) = 0 for negative n), where 3, 5, 14, 18, ... is the sequence A074378. - Ludovic Schwob, Aug 04 2021
Comments