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A208856 Partitions of n into parts not congruent to 0, +-4, +-6, +-10, 16 (mod 32).

%I A208856 #29 Feb 16 2025 08:33:16
%S A208856 1,1,2,3,4,6,8,11,15,20,26,34,44,56,72,91,114,143,178,220,272,334,408,
%T A208856 498,605,732,884,1064,1276,1528,1824,2171,2580,3058,3616,4269,5028,
%U A208856 5910,6936,8124,9498,11088,12922,15034,17468,20264,23472,27154,31369,36189
%N A208856 Partitions of n into parts not congruent to 0, +-4, +-6, +-10, 16 (mod 32).
%C A208856 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A208856 Andrews (1987) refers to this sequence as p(T, n) where T is the set in equation (1) on page 437.
%H A208856 G. C. Greubel, <a href="/A208856/b208856.txt">Table of n, a(n) for n = 0..1000</a>
%H A208856 G. E. Andrews, <a href="http://www.jstor.org/stable/2322727">Unsolved Problems: Further Problems on Partitions</a>, Amer. Math. Monthly 94 (1987), no. 5, 437-439.
%H A208856 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A208856 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A208856 Expansion of (f(x) / f(-x) - 1) / (2 * x) in powers of x where f() is a Ramanujan theta function.
%F A208856 Expansion of (f(x^14, x^34) - x^4 * f(x^2, x^46)) / f(-x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function.
%F A208856 Euler transform of period 32 sequence [ 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, ...].
%F A208856 a(n) = A115671(n + 1). 2 * a(n) = A080054(n + 1). a(2*n) = A187154(n). a(2*n + 1) = A208851(n).
%e A208856 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 11*x^7 + 15*x^8 + 20*x^9 + ...
%e A208856 a(5) = 6 since  5 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 in 6 ways.
%e A208856 a(6) = 8 since  5 + 1 = 3 + 3 = 3 + 2 + 1 = 3 + 1 + 1 + 1 = 2 + 2 + 2 = 2 + 2 + 1 + 1 = 2 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1 in 8 ways.
%t A208856 A208856[n_] := SeriesCoefficient[(1/(2*q))*((QPochhammer[-q, -q]/ QPochhammer[q, q]) - 1), {q, 0, n}]; Table[A208856[n], {n,0,50}] (* _G. C. Greubel_, Jun 19 2017 *)
%o A208856 (PARI) {a(n) = local(A); if( n<0, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)) - 1) / 2, n))}
%Y A208856 Cf. A080054, A115671, A187154, A208851.
%K A208856 nonn
%O A208856 0,3
%A A208856 _Michael Somos_, Mar 02 2012