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A266575 Expansion of q * f(-q^4)^6 / phi(-q) in powers of q where phi(), f() are Ramanujan theta functions.

%I A266575 #20 Feb 16 2025 08:33:28
%S A266575 1,2,4,8,8,12,16,16,25,28,28,32,40,40,48,64,48,62,76,64,80,92,80,96,
%T A266575 121,100,112,128,120,136,160,128,144,184,152,200,200,164,208,224,192,
%U A266575 216,252,224,248,296,224,256,337,262,312,320,280,336,368,320,336,396
%N A266575 Expansion of q * f(-q^4)^6 / phi(-q) in powers of q where phi(), f() are Ramanujan theta functions.
%C A266575 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A266575 G. C. Greubel, <a href="/A266575/b266575.txt">Table of n, a(n) for n = 1..2500</a>
%H A266575 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A266575 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A266575 Expansion of q * phi(q) * psi(q^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions.
%F A266575 Expansion of eta(q^2) * eta(q^4)^6 / eta(q)^2 in powers of q.
%F A266575 Euler transform of period 4 sequence [2, 1, 2, -5, ...].
%F A266575 G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^(-3/2) (t/I)^(5/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A245643.
%F A266575 G.f.: x * Product_{k>0} (1 + x^k) * (1 - x^(4*k))^6 / (1 - x^k).
%F A266575 Convolution inverse of A134414.
%e A266575 G.f. = x + 2*x^2 + 4*x^3 + 8*x^4 + 8*x^5 + 12*x^6 + 16*x^7 + 16*x^8 + ...
%t A266575 a[ n_] := SeriesCoefficient[ q QPochhammer[ q^4]^6 / EllipticTheta[ 4, 0, q], {q, 0, n}];
%t A266575 a[ n_] := SeriesCoefficient[ 2^-4 EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q]^4, {q, 0, n}];
%o A266575 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^6 / eta(x + A)^2, n))};
%o A266575 (Magma) A := Basis( ModularForms( Gamma1(4), 5/2), 59); A[2];
%Y A266575 Cf. A134414. A245643.
%K A266575 nonn
%O A266575 1,2
%A A266575 _Michael Somos_, Jan 03 2016