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A213627 Expansion of psi(x)^4 / psi(x^3) in powers of x where psi() is a Ramanujan theta function.

%I A213627 #19 Feb 16 2025 08:33:17
%S A213627 1,4,6,7,9,6,7,15,12,12,13,6,12,18,18,13,15,18,12,24,12,13,27,12,24,
%T A213627 15,12,24,28,30,12,27,18,12,30,18,19,27,24,24,27,24,36,30,18,19,24,24,
%U A213627 24,45,18,12,45,30,24,28,18,36,36,36,24,15,36,36,51,18,25
%N A213627 Expansion of psi(x)^4 / psi(x^3) in powers of x where psi() is a Ramanujan theta function.
%C A213627 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A213627 Alois P. Heinz, <a href="/A213627/b213627.txt">Table of n, a(n) for n = 0..10000</a> (first 2501 terms from G. C. Greubel)
%H A213627 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A213627 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A213627 Expansion of q^(-1/8) * eta(q^2)^8 * eta(q^3) / (eta(q)^4 * eta(q^6)^2) in powers of q.
%F A213627 a(3*n + 2) = 6 * A212907(n).
%F A213627 Euler transform of period 6 sequence [4, -4, 3, -4, 4, -3, ...]. - _Georg Fischer_, Aug 18 2020
%e A213627 G.f. = 1 + 4*x + 6*x^2 + 7*x^3 + 9*x^4 + 6*x^5 + 7*x^6 + 15*x^7 + 12*x^8 + ...
%e A213627 G.f. = q + 4*q^9 + 6*q^17 + 7*q^25 + 9*q^33 + 6*q^41 + 7*q^49 + 15*q^57 + 12*q^65 + ...
%p A213627 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
%p A213627       add([-3, 4, -4, 3, -4, 4][1+irem(d, 6)]*d,
%p A213627         d=numtheory[divisors](j)), j=1..n)/n)
%p A213627     end:
%p A213627 seq(a(n), n=0..100);  # _Alois P. Heinz_, Aug 18 2020
%t A213627 a[ n_] := SeriesCoefficient[ 1/8 EllipticTheta[ 2, 0, q]^4 / EllipticTheta[ 2, 0, q^3], {q, 0, 2 n + 1/4}];
%o A213627 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^3 + A) / (eta(x + A)^4 * eta(x^6 + A)^2), n))};
%Y A213627 Cf. A212907.
%K A213627 nonn
%O A213627 0,2
%A A213627 _Michael Somos_, Jun 16 2012