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A245432 Expansion of f(-q^3, -q^5)^2 / (psi(-q) * phi(q^2)) in powers of q where phi(), psi(), f() are Ramanujan theta functions.

%I A245432 #11 Feb 16 2025 08:33:23
%S A245432 1,1,-1,-2,3,4,-6,-8,11,15,-20,-26,34,44,-56,-72,91,114,-143,-178,220,
%T A245432 272,-334,-408,498,605,-732,-884,1064,1276,-1528,-1824,2171,2580,
%U A245432 -3058,-3616,4269,5028,-5910,-6936,8124,9498,-11088,-12922,15034,17468,-20264
%N A245432 Expansion of f(-q^3, -q^5)^2 / (psi(-q) * phi(q^2)) in powers of q where phi(), psi(), f() are Ramanujan theta functions.
%C A245432 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A245432 G. C. Greubel, <a href="/A245432/b245432.txt">Table of n, a(n) for n = 0..1000</a>
%H A245432 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A245432 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A245432 Expansion of (f(-q^3, -q^5) / f(-q^1, -q^7)) * (psi(q^4) / phi(q^2)) in powers of q where phi(), psi(), f() are Ramanujan theta functions.
%F A245432 Euler transform of period 8 sequence [ 1, -2, -1, 4, -1, -2, 1, 0, ...].
%F A245432 Convolution quotient of A244526 and A226192.
%F A245432 a(n) = (-1)^floor(n/2) * A115671(n).
%F A245432 a(n) = A224216(n) unless n=0. a(2*n+1) = A210063(n).
%e A245432 G.f. = 1 + q - q^2 - 2*q^3 + 3*q^4 + 4*q^5 - 6*q^6 - 8*q^7 + 11*q^8 + ...
%t A245432 a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2, q^4] / QPochhammer[ q^4, q^8]^2)^2 QPochhammer[ q^3, q^8] QPochhammer[ q^5, q^8] / (QPochhammer[ q^1, q^8] QPochhammer[ q^7, q^8]), {q, 0, n}];
%o A245432 (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[0, -1, 2, 1, -4, 1, 2, -1][k%8 + 1]), n))};
%o A245432 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); (-1)^(n \ 2) * polcoeff( (1 + eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A))) / 2, n))};
%Y A245432 Cf. A115671, A210063, A226192, A224216, A244526.
%K A245432 sign
%O A245432 0,4
%A A245432 _Michael Somos_, Jul 21 2014