cp's OEIS Frontend

A258832 Expansion of psi(-x^3) * f(-x, x^2) in powers of x where psi(), f(,) are Ramanujan theta functions.

%I A258832 #11 Feb 16 2025 08:33:25
%S A258832 1,-1,1,-1,1,-2,0,-1,1,-1,2,-1,1,0,1,-2,1,0,2,-1,1,-1,1,-1,1,-2,1,0,0,
%T A258832 -1,2,-2,1,-1,0,-3,0,-1,1,0,2,0,1,-1,2,-2,1,-1,0,-1,1,-1,2,-1,1,0,1,
%U A258832 -2,1,0,3,0,0,-1,1,-2,1,-1,1,-1,3,-1,0,-1,0,-2,0
%N A258832 Expansion of psi(-x^3) * f(-x, x^2) in powers of x where psi(), f(,) are Ramanujan theta functions.
%C A258832 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A258832 G. C. Greubel, <a href="/A258832/b258832.txt">Table of n, a(n) for n = 0..1000</a>
%H A258832 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A258832 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A258832 Expansion of q^(-5/12) * eta(q) * eta(q^4) * eta(q^6)^4 / (eta(q^2)^2 * eta(q^3) * eta(q^12)) in powers of q.
%F A258832 Euler transform of period 12 sequence [ -1, 1, 0, 0, -1, -2, -1, 0, 0, 1, -1, -2, ...].
%F A258832 G.f.: Product_{k>0} (1 + x^k)^2 * (1 - x^(3*k))^2 * (1 - x^k + x^(2*k))^3 / (1 - x^(2*k) + x^(4*k)).
%F A258832 a(n) = (-1)^n * A121444(n). Convolution square is A258831.
%e A258832 G.f. = 1 - x + x^2 - x^3 + x^4 - 2*x^5 - x^7 + x^8 - x^9 + 2*x^10 + ...
%e A258832 G.f. = q^5 - q^17 + q^29 - q^41 + q^53 - 2*q^65 - q^89 + q^101 - q^113 + ...
%t A258832 a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 12 n + 5, KroneckerSymbol[ -4, #] &] / 2];
%t A258832 a[ n_] := SeriesCoefficient[ QPochhammer[ x^6]^2 QPochhammer[ x, -x] / QPochhammer[ x^3, -x^3], {x, 0, n}];
%o A258832 (PARI) {a(n) = if( n<0, 0, (-1)^n * sumdiv( 12*n + 5, d, kronecker( -4, d)) / 2)};
%o A258832 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^4 / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A)), n))};
%Y A258832 Cf. A121444, A258831.
%K A258832 sign
%O A258832 0,6
%A A258832 _Michael Somos_, Jun 11 2015