A214130 Partitions of n into parts congruent to +-2, +-3 (mod 13).

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 7, 9, 9, 11, 12, 14, 15, 18, 19, 23, 24, 28, 30, 35, 37, 43, 46, 52, 56, 64, 68, 77, 84, 93, 101, 113, 121, 135, 146, 161, 174, 193, 207, 229, 247, 272, 292, 322, 346, 379, 408, 446, 479, 524, 562, 613, 659

Offset: 0

Author

Michael Somos, Jul 04 2012

Keywords

• nonn

Comments

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

Examples

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

Links

• Alois P. Heinz, 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. A214129, A214131.

Programs

Maple

with (numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(irem(d, 13) in [2, 3, 10, 11], d, 0),
          d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 23 2013

Mathematica

a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ q^2, q^13] QPochhammer[ q^3, q^13] QPochhammer[ q^10, q^13] QPochhammer[ q^11, q^13]), {q, 0, n}]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[If[MemberQ[{2, 3, 10, 11}, Mod[d, 13]], d, 0], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

PARI

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

Formula

Expansion of f(-x^13)^2 / (f(-x^2, -x^11) * f(-x^3, -x^10)) in powers of x where f() is Ramanujan's two-variable theta function.

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

G.f.: 1 / (Product_{k>0} (1 - x^(13*k - 2)) * (1 - x^(13*k - 3)) * (1 - x^(13*k - 10)) * (1 - x^(13*k - 11))).

A214129(n) = a(n) + A214131(n-1).