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A261119 Expansion of f(x^2, -x^4) * f(x, x^5)^2 / f(-x^12, -x^12) in powers of x where f(, ) is Ramanujan's general theta function.

%I A261119 #15 Feb 16 2025 08:33:26
%S A261119 1,2,2,2,0,0,1,2,4,0,0,0,0,4,2,2,0,0,3,2,2,2,0,0,2,2,2,0,0,0,0,2,2,2,
%T A261119 0,0,3,2,4,2,0,0,0,6,2,0,0,0,0,2,4,0,0,0,2,2,2,4,0,0,1,0,2,0,0,0,0,2,
%U A261119 6,2,0,0,2,4,0,2,0,0,4,4,0,0,0,0,0,4,2
%N A261119 Expansion of f(x^2, -x^4) * f(x, x^5)^2 / f(-x^12, -x^12) in powers of x where f(, ) is Ramanujan's general theta function.
%C A261119 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A261119 G. C. Greubel, <a href="/A261119/b261119.txt">Table of n, a(n) for n = 0..10000</a>
%H A261119 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A261119 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A261119 Expansion of f(x^2, x^6) * f(x, x^5)^2 / f(x^4, x^8) in powers of x where f(,) is Ramanujan'sgeneral theta function.
%F A261119 Expansion of q^(-3/4) * eta(q^2)^3 * eta(q^3)^2 * eta(q^4) * eta(q^24) / (eta(q)^2 * eta(q^6)^2 * eta(q^8)) in powers of q.
%F A261119 Euler transform of period 24 sequence [ 2, -1, 0, -2, 2, -1, 2, -1, 0, -1, 2, -2, 2, -1, 0, -1, 2, -1, 2, -2, 0, -1, 2, -2, ...].
%F A261119 a(n) = (-1)^n * A257921(n) = A129402(2*n + 1) = A261118(3*n + 2) = A192013(4*n + 3) = A000377(4*n + 3).
%F A261119 a(2*n) = A257920(n). a(2*n + 1) = 2 * A259896(n). a(3*n) = A261118(n).
%e A261119 G.f. = 1 + 2*x + 2*x^2 + 2*x^3 + x^6 + 2*x^7 + 4*x^8 + 4*x^13 + 2*x^14 + ...
%e A261119 G.f. = q^3 + 2*q^7 + 2*q^11 + 2*q^15 + q^27 + 2*q^31 + 4*q^35 + 4*q^55 + ...
%t A261119 a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 3}, (-1)^n DivisorSum[ m, KroneckerSymbol[ 12, #] KroneckerSymbol[ -2, m/#] &]]]; (* _Michael Somos_, Dec 22 2016 *)
%o A261119 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^24 + A) / (eta(x + A)^2 * eta(x^6 + A)^2 * eta(x^8 + A)), n))};
%o A261119 (PARI) a(n) = my(m = 4*n+3); (-1)^n*sumdiv(m, d, kronecker(12, d) * kronecker(-2, m/d)); \\ _Michel Marcus_, Dec 13 2017
%Y A261119 Cf. A000377, A129402, A192013, A257920, A257921, A259896, A261118.
%K A261119 nonn
%O A261119 0,2
%A A261119 _Michael Somos_, Aug 08 2015