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

%I #18 Aug 18 2020 12:58:25

%S 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 15,12,24,28,30,12,27,18,12,30,18,19,27,24,24,27,24,36,30,18,19,24,24,

%U 24,45,18,12,45,30,24,28,18,36,36,36,24,15,36,36,51,18,25

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

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H 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 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of q^(-1/8) * eta(q^2)^8 * eta(q^3) / (eta(q)^4 * eta(q^6)^2) in powers of q.

%F a(3*n + 2) = 6 * A212907(n).

%F Euler transform of period 6 sequence [4, -4, 3, -4, 4, -3, ...]. - _Georg Fischer_, Aug 18 2020

%e 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 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 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*

%p add([-3, 4, -4, 3, -4, 4][1+irem(d, 6)]*d,

%p d=numtheory[divisors](j)), j=1..n)/n)

%p end:

%p seq(a(n), n=0..100); # _Alois P. Heinz_, Aug 18 2020

%t a[ n_] := SeriesCoefficient[ 1/8 EllipticTheta[ 2, 0, q]^4 / EllipticTheta[ 2, 0, q^3], {q, 0, 2 n + 1/4}];

%o (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 Cf. A212907.

%K nonn

%O 0,2

%A _Michael Somos_, Jun 16 2012

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