A214131 Partitions of n into parts congruent to +-4, +-6 (mod 13).

1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 10, 10, 12, 12, 15, 15, 18, 19, 22, 23, 27, 28, 32, 35, 39, 41, 47, 50, 56, 60, 67, 71, 80, 85, 94, 101, 113, 119, 132, 141, 156, 166, 183, 195, 215, 229, 250, 268, 293, 313, 341

Offset

0,13

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^4, -x^9) * f(-x^6, -x^7)) in powers of x where f() is Ramanujan's two-variable theta function.

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

G.f.: 1 / (Product_{k>0} (1 - x^(13*k - 4)) * (1 - x^(13*k - 6)) * (1 - x^(13*k - 7)) * (1 - x^(13*k - 9))).

A214129(n) = A214130(n) + a(n-1).

Example

1 + x^4 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + 2*x^12 + 2*x^13 + 2*x^14 + ...
q^5 + q^29 + q^41 + q^47 + q^53 + q^59 + q^65 + q^71 + 2*q^77 + 2*q^83 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ q^4, q^13] QPochhammer[ q^6, q^13] QPochhammer[ q^7, q^13] QPochhammer[ q^9, q^13]), {q, 0, n}]

Prog

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

Crossrefs

Cf. A214129, A214130.

Sequence in context: A053266 A112217 A172033 * A229977 A094994 A026839

Adjacent sequences: A214128 A214129 A214130 * A214132 A214133 A214134

Keyword

nonn

Author

Michael Somos, Jul 04 2012

Status

approved