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A259660 Expansion of f(-x, -x^11) * psi(-x^3)^2 / psi(-x) in powers of x where psi(), f() are Ramanujan theta functions.

%I A259660 #15 Feb 16 2025 08:33:26
%S A259660 1,0,0,-1,1,1,0,0,1,0,0,-1,1,0,0,1,1,-1,0,-1,2,1,0,0,0,1,0,-1,1,0,0,
%T A259660 -1,1,0,0,0,1,1,0,0,2,0,0,-1,1,0,0,0,1,-1,0,-1,0,1,0,2,1,0,0,-2,2,0,0,
%U A259660 0,1,1,0,-1,0,1,0,0,2,-1,0,0,1,0,0,0,1,-1
%N A259660 Expansion of f(-x, -x^11) * psi(-x^3)^2 / psi(-x) in powers of x where psi(), f() are Ramanujan theta functions.
%C A259660 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A259660 G. C. Greubel, <a href="/A259660/b259660.txt">Table of n, a(n) for n = 0..1000</a>
%H A259660 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A259660 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A259660 Expansion of f(-x, -x^11) * f(x, x^5)^2 / f(x) in powers of x where f(,) is the Ramanujan general theta function.
%F A259660 Euler transform of period 12 sequence [ 0, 0, -1, 1, 1, 0, 1, 1, -1, 0, 0, -2, ...].
%F A259660 a(4*n) = A121444(n). a(4*n + 1) = a(n - 1). a(4*n + 2) = 0.
%F A259660 Convolution of A247133 and A259529.
%e A259660 G.f. = 1 - x^3 + x^4 + x^5 + x^8 - x^11 + x^12 + x^15 + x^16 - x^17 - x^19 + ...
%e A259660 G.f. = q^5 - q^14 + q^17 + q^20 + q^29 - q^38 + q^41 + q^50 + q^53 - q^56 + ...
%t A259660 a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 0, 2, 0, 0, 1, -1, -1, 0, -1, -1, 1, 0}[[Mod[ k, 12, 1]]], {k, n}], {x, 0, n}];
%t A259660 QP:= QPochhammer; a[n_]:= SeriesCoefficient[(QP[x, x^12]*QP[x^11,x^12]* QP[x^12]*QP[x^3, -x^3]^2*QP[x^6]^2)/(QP[x, -x]*QP[x^2]), {x, 0, n}]; Table[a[n], {n, 0, 100}] (* _G. C. Greubel_, Mar 17 2018 *)
%o A259660 (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^([ 2, 0, 0, 1, -1, -1, 0, -1, -1, 1, 0, 0][k%12 + 1]), 1 + x * O(x^n)), n))};
%Y A259660 Cf. A121444, A247133, A259529.
%K A259660 sign
%O A259660 0,21
%A A259660 _Michael Somos_, Jul 02 2015