cp's OEIS Frontend

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

%I A214130 #21 Feb 16 2025 08:33:17
%S A214130 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,
%T A214130 30,35,37,43,46,52,56,64,68,77,84,93,101,113,121,135,146,161,174,193,
%U A214130 207,229,247,272,292,322,346,379,408,446,479,524,562,613,659
%N A214130 Partitions of n into parts congruent to +-2, +-3 (mod 13).
%C A214130 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A214130 Alois P. Heinz, <a href="/A214130/b214130.txt">Table of n, a(n) for n = 0..1000</a>
%H A214130 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A214130 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A214130 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.
%F A214130 Euler transform of period 13 sequence [ 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, ...].
%F A214130 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))).
%F A214130 A214129(n) = a(n) + A214131(n-1).
%e A214130 1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 3*x^10 + ...
%e A214130 q^-1 + q^11 + q^17 + q^23 + q^29 + 2*q^35 + q^41 + 2*q^47 + 2*q^53 + ...
%p A214130 with (numtheory):
%p A214130 a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A214130       `if`(irem(d, 13) in [2, 3, 10, 11], d, 0),
%p A214130           d=divisors(j)) *a(n-j), j=1..n)/n)
%p A214130     end:
%p A214130 seq(a(n), n=0..100);  # _Alois P. Heinz_, Oct 23 2013
%t A214130 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}]
%t A214130 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_ *)
%o A214130 (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))}
%Y A214130 Cf. A214129, A214131.
%K A214130 nonn
%O A214130 0,7
%A A214130 _Michael Somos_, Jul 04 2012