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A208851 Partitions of 2*n + 1 into parts not congruent to 0, +-4, +-6, +-10, 16 (mod 32).

%I A208851 #22 Feb 16 2025 08:33:16
%S A208851 1,3,6,11,20,34,56,91,143,220,334,498,732,1064,1528,2171,3058,4269,
%T A208851 5910,8124,11088,15034,20264,27154,36189,47988,63324,83176,108780,
%U A208851 141672,183776,237499,305812,392406,501856,639781,813108,1030354,1301928,1640572,2061850
%N A208851 Partitions of 2*n + 1 into parts not congruent to 0, +-4, +-6, +-10, 16 (mod 32).
%C A208851 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700)
%H A208851 G. C. Greubel, <a href="/A208851/b208851.txt">Table of n, a(n) for n = 0..1000</a>
%H A208851 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A208851 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A208851 Expansion of (phi(q^2) / phi(-q) - 1) / (2 * q) in powers of q where phi() is a Ramanujan theta function.
%F A208851 Euler transform of period 16 sequence [ 3, 0, 1, 2, 1, 2, 3, 0, 3, 2, 1, 2, 1, 0, 3, 0, ...].
%F A208851 2 * a(n) = A208850(n + 1). a(n) = A185083(n + 1).
%e A208851 1 + 3*q + 6*q^2 + 11*q^3 + 20*q^4 + 34*q^5 + 56*q^6 + 91*q^7 + 143*q^8 + ...
%e A208851 a(2) = 6 since  2*2 + 1 = 5 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 in 6 ways.
%t A208851 a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q^2]/EllipticTheta[3, 0, -q] - 1)/(2*q), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Mar 05 2018 *)
%o A208851 (PARI) {a(n) = local(A); if( n<0, 0, n = 2*n + 2; A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)) - 1) / 2, n))}
%Y A208851 Cf. A185083, A208850.
%K A208851 nonn
%O A208851 0,2
%A A208851 _Michael Somos_, Mar 02 2012