cp's OEIS Frontend

A258939 Expansion of f(-x^3, -x^5) * f(x^3, x^13) / (f(-x, -x^2) * f(-x^8, -x^16)) in powers of x where f(, ) is Ramanujan's general theta function.

%I A258939 #11 Feb 16 2025 08:33:25
%S A258939 1,1,2,3,5,6,9,12,17,22,30,38,51,64,83,104,133,165,208,256,319,390,
%T A258939 481,584,715,863,1047,1258,1517,1812,2172,2584,3080,3648,4327,5104,
%U A258939 6028,7084,8330,9756,11430,13340,15574,18122,21086,24464,28378,32832,37977,43823
%N A258939 Expansion of f(-x^3, -x^5) * f(x^3, x^13) / (f(-x, -x^2) * f(-x^8, -x^16)) in powers of x where f(, ) is Ramanujan's general theta function.
%C A258939 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A258939 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A258939 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A258939 Euler transform of period 32 sequence [ 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, ...].
%F A258939 - a(n) = A029838(4*n + 2).
%F A258939 a(n) ~ sqrt(2*(1+sqrt(2))) * exp(Pi*sqrt(n/2)) / (16*n^(3/4)). - _Vaclav Kotesovec_, Nov 07 2015
%e A258939 G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 17*x^8 + ...
%e A258939 G.f. = q^15 + q^47 + 2*q^79 + 3*q^111 + 5*q^143 + 6*q^175 + 9*q^207 + ...
%t A258939 a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^-{ 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0}[[Mod[k, 32, 1]]], {k, n}], {x, 0, n}];
%o A258939 (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^-[ 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1][k%32 + 1]), n))};
%Y A258939 Cf. A029838.
%K A258939 nonn
%O A258939 0,3
%A A258939 _Michael Somos_, Nov 07 2015