OFFSET
0,2
COMMENTS
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with odd parts distinct, Ramanujan J. 22 (2010), 273--284.
Michael Somos, Introduction to Ramanujan theta functions
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.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
G.f.: Product_{k>=1} (1 + x^k)^2 / (1 - x^(4*k))^2, corrected by Vaclav Kotesovec, Mar 25 2017
Expansion of 1 / psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.
a(n) ~ exp(Pi*sqrt(n))/(2^(5/2)*n^(5/4)). - Vaclav Kotesovec, Jul 05 2016
Euler transform of period 4 sequence [2, 0, 2, 2, ...]. - Michael Somos, Mar 02 2019
EXAMPLE
a(4)=11 because "(0,4)=(0,3+1)=(0,2+2)=(1,3)=(1,2+1)=(2,2)=(4,0)=(3+1,0)=(2+2,0)=(3,1)=(2+1,1)".
G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 11*x^4 + 18*x^5 + 28*x^6 + 44*x^7 + ... - Michael Somos, Mar 02 2019
G.f. = q^-1 + 2*q^3 + 3*q^7 + 6*q^11 + 11*q^15 + 18*q^19 + 28*q^23 + ... - Michael Somos, Mar 02 2019
MAPLE
Digits:=200:with(PolynomialTools): with(qseries): with(ListTools):
GenFun:=series(etaq(q, 2, 100)^2/etaq(q, 1, 100)^2/etaq(q, 4, 100)^2, q, 50):
CoefficientList(sort(convert(GenFun, polynom), q, ascending), q);
MATHEMATICA
s = QPochhammer[-1, x]^2/(4*QPochhammer[x^4, x^4]^2) + O[x]^40; CoefficientList[s, x] (* Jean-François Alcover, May 20 2016 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2, x^4] / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, Mar 02 2019 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0 , A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A))^2, n))}; /* Michael Somos, Mar 02 2019 */
CROSSREFS
KEYWORD
nonn
AUTHOR
M.S. Mahadeva Naika, May 18 2016
STATUS
approved