A185083 Partitions of 2*n into parts not congruent to 0, +-2, +-12, +-14, 16 (mod 32).

1, 1, 3, 6, 11, 20, 34, 56, 91, 143, 220, 334, 498, 732, 1064, 1528, 2171, 3058, 4269, 5910, 8124, 11088, 15034, 20264, 27154, 36189, 47988, 63324, 83176, 108780, 141672, 183776, 237499, 305812, 392406, 501856, 639781, 813108, 1030354, 1301928, 1640572

Offset: 0

Author

Michael Somos, Mar 02 2012

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			1 + x + 3*x^2 + 6*x^3 + 11*x^4 + 20*x^5 + 34*x^6 + 56*x^7 + 91*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A115671, A208850, A208851.

Programs

Mathematica

f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A185083[n_] := SeriesCoefficient[(1/2)*(f[x^2, x^2]/f[-x, -x] + 1), {x, 0, n}]; Table[A185083[n], {n,0,50}] (* G. C. Greubel, Jun 22 2017 *)

PARI

{a(n) = local(A); if( n<0, 0, n = 2*n; A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)) + 1) / 2, n))}

Formula

Expansion of (phi(q^2) / phi(-q) + 1) / 2 in powers of q where phi() is a Ramanujan theta function.

Euler transform of period 16 sequence [ 1, 2, 3, 2, 3, 0, 1, 0, 1, 0, 3, 2, 3, 2, 1, 0, ...].

2 * a(n) = A208850(n) unless n = 0. a(n + 1) = A208851(n). a(n) = A115671(2*n).