OFFSET
0,7
COMMENTS
LINKS
Cristina Ballantine and Mircea Merca, 6-regular partitions: new combinatorial properties, congruences, and linear inequalities, arXiv:2302.01253 [math.NT], 2023.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-5/24) * (eta(q^2)^2 * eta(q^12)) / (eta(q) * eta(q^4) * eta(q^6)) in power of q.
Euler transform of period 12 sequence [1, -1, 1, 0, 1, 0, 1, 0, 1, -1, 1, 0, ...].
G.f.: Product_{k>=1} (1 + x^(6*k))/(1 + (-x)^k) = Product_{k>=1} (1 + x^(2*k-1)) * (1 + x^(6*k)).
A261736(n) = (-1)^n * a(n).
a(n) ~ exp(sqrt(2*n)*Pi/3) / (2^(7/4)*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Oct 31 2019
EXAMPLE
G.f. = 1 + x + x^3 + x^4 + x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 3*x^9 + ...
G.f. = q^5 + q^29 + q^77 + q^101 + q^125 + 2*q^149 + 2*q^173 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^6, x^6], {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^12 + A)) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 27 2019
STATUS
approved