A214129 Partitions of n into parts congruent to +-1, +-5 (mod 13).

1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 17, 19, 21, 24, 27, 31, 34, 38, 42, 47, 52, 58, 64, 71, 78, 87, 95, 105, 116, 128, 140, 154, 168, 185, 202, 221, 241, 264, 287, 314, 341, 373, 405, 441, 478, 520, 564, 612, 662, 719, 777, 842

Offset

0,6

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 f(-x^13)^2 / (f(-x, -x^12) * f(-x^5, -x^8)) in powers of x where f() is Ramanujan's two-variable theta function.

Euler transform of period 13 sequence [ 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, ...].

G.f.: 1 / (Product_{k>0} (1 - x^(13*k - 1)) * (1 - x^(13*k - 5)) * (1 - x^(13*k - 8)) * (1 - x^(13*k - 12))).

a(n) = A214130(n) + A214131(n-1).

Example

1 + x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + ...
q^-1 + q^5 + q^11 + q^17 + q^23 + 2*q^29 + 2*q^35 + 2*q^41 + 3*q^47 + 3*q^53 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ q, q^13] QPochhammer[ q^5, q^13] QPochhammer[ q^8, q^13] QPochhammer[ q^12, q^13]), {q, 0, n}]

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - [ 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1][k%13 + 1] * x^k, 1 + x * O(x^n)), n))}

Crossrefs

Cf. A214130, A214131.

Sequence in context: A280169 A018117 A086936 * A280151 A094990 A120178

Adjacent sequences: A214126 A214127 A214128 * A214130 A214131 A214132

Keyword

nonn

Author

Michael Somos, Jul 04 2012

Status

approved