OFFSET
0,4
COMMENTS
Also number of partitions of n into distinct parts in which no part is multiple of 4. - Vladeta Jovovic, Jul 31 2004
McKay-Thompson series of class 64a for the Monster group.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..220 from Reinhard Zumkeller)
G. E. Andrews and R. P. Lewis, An algebraic identity of F. H. Jackson and its implications for partitions, Discrete Math., 232 (2001), 77-83.
Cristina Ballantine and Mircea Merca, 4-Regular partitions and the pod function, arXiv:2111.10702 [math.CO], 2021.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
M. D. Hirschhorn, J. A. Sellers, A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014).
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 12.
FORMULA
G.f.: Product_{i>0} (1+x^i)/(1+x^(4*i)). - Vladeta Jovovic, Jul 31 2004
Expansion of chi(x) * chi(x^2) = psi(x) / psi(-x^2) = phi(-x^4) / psi(-x) = chi(-x^4) / chi(-x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions. - Michael Somos, Jul 01 2014
Expansion of q^(1/8) * eta(q^2) * eta(q^4) / (eta(q) * eta(q^8)) in powers of q.
Euler transform of period 8 sequence [1, 0, 1, -1, 1, 0, 1, 0, ...].
Given g.f. A(x), then B(q) = A(q^8) / q satisifes 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) + 3*u*v. - Michael Somos, Jul 01 2014
G.f.: Product_{k>0} (1 - x^(8*k - 4)) / (1 - x^(2*k - 1)).
a(n) ~ exp(sqrt(n)*Pi/2) / (4*n^(3/4)) * (1 - (3/(4*Pi) + Pi/32) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017
EXAMPLE
G.f. = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + ...
T64a = 1/q + q^7 + q^15 + 2*q^23 + q^31 + 2*q^39 + 3*q^47 + 3*q^55 + 4*q^63 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^2, x^4]), {x, 0, n}]; (* Michael Somos, Jul 01 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^4] / (QPochhammer[ x] QPochhammer[ x^8]), {x, 0, n}]; (* Michael Somos, Jul 01 2014 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0 , A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A) / (eta(x + A) * eta(x^8 + A)), n))};
(Haskell)
a070048 = p a042968_list where
p _ 0 = 1
p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Oct 01 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, May 09 2002
EXTENSIONS
Additional comments from Michael Somos, Dec 04 2002
STATUS
approved