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A300574 Coefficient of x^n in 1/((1-x)(1+x^3)(1-x^5)(1+x^7)(1-x^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 (list; graph; refs; listen; history; text; internal format)
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*k-1))*(1-x^(4*k-3))), {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

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Last modified September 27 11:57 EDT 2021. Contains 347689 sequences. (Running on oeis4.)