A208851 Partitions of 2*n + 1 into parts not congruent to 0, +-4, +-6, +-10, 16 (mod 32).

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, 2061850

Offset

0,2

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 * q) in powers of q where phi() is a Ramanujan theta function.

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

2 * a(n) = A208850(n + 1). a(n) = A185083(n + 1).

Example

1 + 3*q + 6*q^2 + 11*q^3 + 20*q^4 + 34*q^5 + 56*q^6 + 91*q^7 + 143*q^8 + ...
a(2) = 6 since  2*2 + 1 = 5 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 in 6 ways.

Mathematica

a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q^2]/EllipticTheta[3, 0, -q] - 1)/(2*q), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 05 2018 *)

Prog

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

Crossrefs

Cf. A185083, A208850.

Sequence in context: A116365 A297443 A185083 * A182845 A265076 A055417

Adjacent sequences: A208848 A208849 A208850 * A208852 A208853 A208854

Keyword

nonn

Author

Michael Somos, Mar 02 2012

Status

approved