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A273228
G.f. is the fourth power of the g.f. of A006950.
2
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
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
Sequence in context: A266367 A316528 A152548 * A291727 A291224 A090855
KEYWORD
nonn
AUTHOR
M.S. Mahadeva Naika, May 18 2016
EXTENSIONS
Edited by N. J. A. Sloane, May 26 2016
STATUS
approved