OFFSET
0,3
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also number of partitions of n such that every part is not congruent to 3 mod 6. More generally, g.f. for number of partitions of n such that every odd part occurs at most m times is Product_{n=1..oo} (1-q^((m+1)*(2*n-1)))/(1-q^n). Similarly, g.f. for number of partitions of n such that every even part occurs at most m times is Product_{n=1..oo} (1-q^((2*m+2)*n))/(1-q^n). - Vladeta Jovovic, Aug 01 2007
LINKS
Brian Drake and Seiichi Manyama, Table of n, a(n) for n = 0..1000 (first 101 terms from Brian Drake)
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
Mircea Merca, Overpartitions in terms of 2-adic valuation, Aequat. Math. (2024). See p. 11.
James A. Sellers, Elementary Proofs of Two Congruences for Partitions with Odd Parts Repeated at Most Twice, arXiv:2409.12321 [math.NT], 2024. See p. 2.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Fabrizio Zanello, Three more proofs of two congruences for Merca's partition function, arXiv:2509.09833 [math.CO], 2025.
FORMULA
G.f.: product_{n=1..oo} (1-q^(6n-3))/(1-q^n).
Expansion of chi(-x^3) / f(-x) in powers of x where chi(), f() are Ramanujan theta functions. - Michael Somos, Aug 05 2007
Expansion of q^(1/6) * eta(q^3) / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Aug 05 2007
Euler transform of period 6 sequence [ 1, 1, 0, 1, 1, 1, ...]. - Michael Somos, Aug 05 2007
a(n) ~ sqrt(5) * exp(sqrt(5*n)*Pi/3) / (12*n). - Vaclav Kotesovec, Dec 11 2016
EXAMPLE
a(6) = 8 because we have 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 2+2+2 and 2+2+1+1. The three excluded partitions of 6 are 3+1+1+1, 2+1+1+1+1 and 1+1+1+1+1+1.
G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 8*x^6 + 10*x^7 + 15*x^8 + ...
G.f. = 1/q + q^5 + 2*q^11 + 2*q^17 + 4*q^23 + 5*q^29 + 8*q^35 + 10*q^41 + ...
MAPLE
A:= series(product( (1-q^(6*n-3))/(1-q^n), n=1..20), q, 21): seq(coeff(A, q, i), i=0..20);
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] / (QPochhammer[ x] QPochhammer[ x^6]), {x, 0, n}]; (* Michael Somos, Jan 29 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^6] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *)
nmax = 50; CoefficientList[Series[Product[1 / ((1-x^k) * (1+x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 11 2016 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)), n))}; /* Michael Somos, Aug 05 2007 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Brian Drake, Jul 30 2007
STATUS
approved