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

%I A214129 #14 Feb 16 2025 08:33:17
%S A214129 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,
%T A214129 38,42,47,52,58,64,71,78,87,95,105,116,128,140,154,168,185,202,221,
%U A214129 241,264,287,314,341,373,405,441,478,520,564,612,662,719,777,842
%N A214129 Partitions of n into parts congruent to +-1, +-5 (mod 13).
%C A214129 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A214129 Evans (1990) page 99 equation (1.10) is the connection between A214129, A214130, A214131. - _Michael Somos_, Nov 01 2024
%H A214129 G. C. Greubel, <a href="/A214129/b214129.txt">Table of n, a(n) for n = 0..1000</a>
%H A214129 Ronald J. Evans, <a href="https://mathweb.ucsd.edu/~revans/Theta.pdf">Theta Function Identities</a>, J. of Mathematical Analysis and Applications 147(1990), 97-121.
%H A214129 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A214129 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A214129 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.
%F A214129 Euler transform of period 13 sequence [ 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, ...].
%F A214129 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))).
%F A214129 a(n) = A214130(n) + A214131(n-1).
%e A214129 G.f. = 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 + ...
%e A214129 G.f. = 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 + ...
%t A214129 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}]
%o A214129 (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))}
%Y A214129 Cf. A214130, A214131.
%K A214129 nonn
%O A214129 0,6
%A A214129 _Michael Somos_, Jul 04 2012