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A185083 Partitions of 2*n into parts not congruent to 0, +-2, +-12, +-14, 16 (mod 32).

%I A185083 #17 Feb 16 2025 08:33:13
%S A185083 1,1,3,6,11,20,34,56,91,143,220,334,498,732,1064,1528,2171,3058,4269,
%T A185083 5910,8124,11088,15034,20264,27154,36189,47988,63324,83176,108780,
%U A185083 141672,183776,237499,305812,392406,501856,639781,813108,1030354,1301928,1640572
%N A185083 Partitions of 2*n into parts not congruent to 0, +-2, +-12, +-14, 16 (mod 32).
%C A185083 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A185083 G. C. Greubel, <a href="/A185083/b185083.txt">Table of n, a(n) for n = 0..1000</a>
%H A185083 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A185083 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A185083 Expansion of (phi(q^2) / phi(-q) + 1) / 2 in powers of q where phi() is a Ramanujan theta function.
%F A185083 Euler transform of period 16 sequence [ 1, 2, 3, 2, 3, 0, 1, 0, 1, 0, 3, 2, 3, 2, 1, 0, ...].
%F A185083 2 * a(n) = A208850(n) unless n = 0. a(n + 1) = A208851(n). a(n) = A115671(2*n).
%e A185083 1 + x + 3*x^2 + 6*x^3 + 11*x^4 + 20*x^5 + 34*x^6 + 56*x^7 + 91*x^8 + ...
%t A185083 f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A185083[n_] := SeriesCoefficient[(1/2)*(f[x^2, x^2]/f[-x, -x] + 1), {x, 0, n}]; Table[A185083[n], {n,0,50}] (* _G. C. Greubel_, Jun 22 2017 *)
%o A185083 (PARI) {a(n) = local(A); if( n<0, 0, n = 2*n; A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)) + 1) / 2, n))}
%Y A185083 Cf. A115671, A208850, A208851.
%K A185083 nonn
%O A185083 0,3
%A A185083 _Michael Somos_, Mar 02 2012