A300575 -id:A300575 - cp's OEIS frontend

A300574 Coefficient of x^n in 1/((1-x)(1+x^3)(1-x^5)(1+x^7)(1-x^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

Offset: 0

Gus Wiseman, Mar 08 2018

By Theorem 1 of Craig, the values a(n) in this list are known to be nonnegative. Combined with Theorem 2 of Seo and Yee, this shows 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|. - William Craig, Dec 31 2021

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.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Gus Wiseman)
Vincent E. Coll, Andrew W. Mayers, and Nick W. Mayers, Statistics on integer partitions arising from seaweed algebras, arXiv:1809.09271 [math.CO], 2018.
William Craig, Seaweed Algebras and the Index Statistic for Partitions, arXiv:2112.09269 [math.CO], 2021.
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000009, A010815, A027193, A067659, A078408, A081362, A099323, A220418, A290261, A292043, A292137, A298118, A300575.

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 *)

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

A193863 Expansion of Product_{n>=0} (1 + q(-q^2)^n) / (1 - q(-q^2)^n).

Original entry on oeis.org

1, 2, 2, 0, -2, 0, 4, 4, -2, -6, 0, 8, 4, -8, -8, 8, 14, -4, -18, 0, 24, 8, -28, -20, 28, 34, -24, -48, 16, 64, 0, -76, -18, 88, 44, -96, -78, 96, 116, -88, -160, 68, 208, -32, -252, -16, 296, 84, -332, -170, 354, 272, -360, -392, 344, 528, -296, -672, 216, 824, -96, -976, -72, 1116, 286, -1240, -552, 1336, 876, -1384

Offset: 0

Joerg Arndt, Aug 07 2011

sign

Expansion of E(-q^2, +q) for E(q,x) = Product_{n>=0} ( 1 + x*q^n ) / ( 1 - x*q^n ).

Replacing q by -q in the g.f. gives the inverse of the g.f., whose expansion is obtained by negating every second term.

			1 + 2*x + 2*x^2 - 2*x^4 + 4*x^6 + 4*x^7 - 2*x^8 - 6*x^9 + 8*x^11 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
J. Arndt, Regularized q-exponential and q-trigonometric series of the first kind

Cf. A015128 E(+q,+q), A002448 E(+q,-q), A000122 E(-q,+q), A004402 E(-q,-q), A080054 E(+q^2,+q), A108494 E(+q^2,-q), A300574, A300575.

N=66; q='q+O('q^N); /* that many terms */
gf = prod(n=0, N, (1+q*(-q^2)^n)/(1-q*(-q^2)^n) );
Vec(gf) /* show terms */
/* Alternative computation of the g.f. using a product form */
V=[0,-2, 1, 2, 0, -2, -1, 2]; /* note vectors are one-based */
gf=prod(n=0, N, (1-q^n)^(V[n%8+1]) );

{a(n) = local(A); if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k + x * O(x^n) )^[ 0, -2, 1, 2, 0, -2, -1, 2][k%8 + 1]), n))} /* Michael Somos, Feb 26 2012 */

Euler transform of period 8 sequence [ 2, -1, -2, 0, 2, 1, -2, 0, ...]. - Michael Somos, Feb 26 2012
G.f.: prod(n>=0, (1+q*(-q^2)^n)/(1-q*(-q^2)^n) ).
G.f.: sum(n>=0, prod(k=0..n-1, 1+(-q^2)^k )/prod(k=1..n, 1-(-q^2)^k ) * q^n ).
G.f.: sum(n>=0, prod(k=0..n-1, 1+(-q^2)^k)/( prod(k=1..n, 1-(-q^2)^k) * prod(k=0..n-1, 1-q*(-q^2)^k ) ) * q^n * (-q^2)^(n*(n-1)/2) ).
Convolution of A300574 and A300575. - Seiichi Manyama, Nov 22 2019

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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)...).

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A193863 Expansion of Product_{n>=0} (1 + q(-q^2)^n) / (1 - q(-q^2)^n).

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cp's OEIS Frontend

A300574 Coefficient of x^n in 1/((1-x)(1+x^3)(1-x^5)(1+x^7)(1-x^9)...).

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A193863 Expansion of Product_{n>=0} (1 + q*(-q^2)^n) / (1 - q*(-q^2)^n).

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Keywords

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PARI

PARI

Formula

A193863 Expansion of Product_{n>=0} (1 + q(-q^2)^n) / (1 - q(-q^2)^n).