A273226 G.f. is the cube of the g.f. of A006950.

1, 3, 6, 13, 27, 51, 91, 159, 273, 455, 738, 1179, 1860, 2886, 4410, 6667, 9981, 14781, 21671, 31512, 45474, 65113, 92547, 130689, 183439, 255930, 355017, 489895, 672672, 919152, 1250107, 1692846, 2282895, 3066180, 4102224, 5468160, 7263217, 9614436, 12684633, 16682276

Offset: 0

Author

M.S. Mahadeva Naika, May 18 2016

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with odd parts distinct, Ramanujan J. 22 (2010), 273--284.
• M. S. Mahadeva Naika and D. S. Gireesh, Arithmetic Properties of Partition k-tuples with Odd Parts Distinct, JIS, Vol. 19 (2016), Article 16.5.7
• 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.

Crossrefs

Cf. A006950.

Programs

Maple

N:= 50:
G:= mul((1+x^k)^3,k=1..N)/mul((1-x^(4*k))^3,k=1..N/4):
S:= series(G,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Jan 21 2019

Mathematica

s = QPochhammer[-1, x]^3/(8*QPochhammer[x^4, x^4]^3) + O[x]^40; CoefficientList[s, x] (* Jean-François Alcover, May 20 2016 *)

Formula

G.f.: Product_{k>=1} (1 + x^k)^3 / (1 - x^(4*k))^3, corrected by Vaclav Kotesovec, Mar 25 2017.

a(n) ~ 3*exp(sqrt(3*n/2)*Pi) / (16*n^(3/2)). - Vaclav Kotesovec, Mar 25 2017

Extensions

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