A273228 G.f. is the fourth power of the g.f. of A006950.

1, 4, 10, 24, 55, 116, 230, 440, 819, 1480, 2602, 4480, 7580, 12604, 20620, 33272, 53029, 83520, 130088, 200600, 306488, 464168, 697150, 1039032, 1537435, 2259300, 3298428, 4785880, 6903657, 9903040, 14129846, 20058488, 28336790, 39845456, 55778050, 77747328, 107924347, 149221160

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with odd parts distinct, Ramanujan J. 22 (2010), 273--284.

L. Wang, Arithmetic properties of partition triples with odd parts distinct, Int. J. Number Theory, 11 (2015), 1791--1805.

L. Wang, Arithmetic properties of partition quadruples with odd parts distinct, Bull. Aust. Math. Soc., doi:10.1017/S0004972715000647.

L. Wang, New congruences for partitions where the odd parts are distinct, J. Integer Seq. (2015), article 15.4.2.

Formula

G.f.: Product_{k>=1} (1 + x^k)^4 / (1 - x^(4*k))^4, corrected by Vaclav Kotesovec, Mar 25 2017

Expansion of 1 / psi(-x)^4 in powers of x where psi() is a Ramanujan theta function.

a(n) ~ exp(sqrt(2*n)*Pi) / (2^(9/4)*n^(7/4)). - Vaclav Kotesovec, Mar 25 2017

Maple

Digits:=200:with(PolynomialTools): with(qseries): with(ListTools):
GenFun:=series(etaq(q, 2, 1000)^4/etaq(q, 1, 1000)^4/etaq(q, 4, 1000)^4, q, 50):
CoefficientList(sort(convert(GenFun, polynom), q, ascending), q);

Mathematica

nmax = 30; CoefficientList[Series[Product[(1 + x^k)^4 / (1 - x^(4*k))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 25 2017 *)
CoefficientList[Series[1/(QPochhammer[q, -q]*QPochhammer[q^2, q^2])^4, {q, 0, 50}], q] (* G. C. Greubel, Apr 17 2018 *)

Crossrefs

Cf. A006960, A273226.

Sequence in context: A266367 A316528 A152548 * A291727 A291224 A090855

Adjacent sequences: A273225 A273226 A273227 * A273229 A273230 A273231

Keyword

nonn

Author

M.S. Mahadeva Naika, May 18 2016

Extensions

Edited by N. J. A. Sloane, May 26 2016

Status

approved