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

0,3

Comments

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

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

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).

Example

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

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 *)

Prog

(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))}

Crossrefs

Cf. A115671, A208850, A208851.

Sequence in context: A180086 A116365 A297443 * A208851 A182845 A265076

Adjacent sequences: A185080 A185081 A185082 * A185084 A185085 A185086

Keyword

nonn

Author

Michael Somos, Mar 02 2012

Status

approved