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

%I A214131 #12 Feb 16 2025 08:33:17
%S A214131 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,
%T A214131 15,18,19,22,23,27,28,32,35,39,41,47,50,56,60,67,71,80,85,94,101,113,
%U A214131 119,132,141,156,166,183,195,215,229,250,268,293,313,341
%N A214131 Partitions of n into parts congruent to +-4, +-6 (mod 13).
%C A214131 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A214131 G. C. Greubel, <a href="/A214131/b214131.txt">Table of n, a(n) for n = 0..1000</a>
%H A214131 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A214131 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A214131 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.
%F A214131 Euler transform of period 13 sequence [ 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, ...].
%F A214131 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))).
%F A214131 A214129(n) = A214130(n) + a(n-1).
%e A214131 1 + x^4 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + 2*x^12 + 2*x^13 + 2*x^14 + ...
%e A214131 q^5 + q^29 + q^41 + q^47 + q^53 + q^59 + q^65 + q^71 + 2*q^77 + 2*q^83 + ...
%t A214131 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}]
%o A214131 (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))}
%Y A214131 Cf. A214129, A214130.
%K A214131 nonn
%O A214131 0,13
%A A214131 _Michael Somos_, Jul 04 2012