

A300574


Coefficient of x^n in 1/((1x)(1+x^3)(1x^5)(1+x^7)(1x^9)...).


9



1, 1, 1, 0, 0, 1, 2, 1, 0, 0, 2, 2, 1, 0, 2, 3, 2, 0, 2, 4, 4, 0, 1, 4, 6, 2, 1, 4, 8, 4, 2, 4, 10, 6, 2, 3, 12, 10, 4, 2, 13, 14, 8, 2, 14, 18, 12, 2, 14, 22, 18, 3, 14, 26, 26, 6, 14, 29, 34, 10, 14, 32, 44, 16, 14, 34, 56, 26, 16, 34, 67, 38, 20, 34, 78, 52, 26
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OFFSET

0,7


COMMENTS

Is this sequence strictly nonnegative? If so, it would be interesting to have a combinatorial interpretation.
Theorem 2 of Seo and Yee seems to show that a(n) = number of partitions of n into odd parts with an odd index minus the number of partitions of n into odd parts with an even index.  N. J. A. Sloane, Feb 21 2020


REFERENCES

Seunghyun Seo and Ae Ja Yee, Index of seaweed algebras and integer partitions, Electronic Journal of Combinatorics, 27:1 (2020), #P1.47. See Conjecture 1 and Theorem 2.


LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Gus Wiseman)
Vincent E. Coll, Andrew W. Mayers, Nick W. Mayers, Statistics on integer partitions arising from seaweed algebras, arXiv:1809.09271 [math.CO], 2018.
Vaclav Kotesovec, Graph  the asymptotic ratio (10000 terms)
Vaclav Kotesovec, Graph  the asymptotic ratio (100000 terms)


FORMULA

O.g.f.: Product_{n >= 0} 1/(1  (1)^n x^(2n+1)).
a(n) = Sum (1)^k where the sum is over all integer partitions of n into odd parts and k is the number of parts not congruent to 1 modulo 4.
a(n) has average order Gamma(1/4) * exp(sqrt(n/3)*Pi/2) / (2^(9/4) * 3^(1/8) * Pi^(3/4) * n^(5/8)).  Vaclav Kotesovec, Jun 04 2019


MATHEMATICA

CoefficientList[Series[1/QPochhammer[x, x^2], {x, 0, 100}], x]
nmax = 100; CoefficientList[Series[Product[1/((1+x^(4*k1))*(1x^(4*k3))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 04 2019 *)


CROSSREFS

Cf. A000009, A010815, A027193, A067659, A078408, A081362, A099323, A220418, A290261, A292043, A292137, A298118, A300575.
Sequence in context: A328795 A055791 A245842 * A327757 A191400 A168315
Adjacent sequences: A300571 A300572 A300573 * A300575 A300576 A300577


KEYWORD

nonn,look,nice


AUTHOR

Gus Wiseman, Mar 08 2018


STATUS

approved



