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A135828 Expansion of psi(x^2)^8 * (psi(x)^8 + psi(-x)^8) / 2 in powers of x^2 where psi() is a Ramanujan theta function.

%I A135828 #16 Feb 16 2025 08:33:07
%S A135828 1,36,378,2200,8955,28836,78558,188568,410805,828080,1564686,2804976,
%T A135828 4809370,7927380,12643560,19594632,29568204,43626708,63094550,
%U A135828 89501040,124916931,171803652,232822908,311683680,412601490,539849556,699657642,898801400,1143680535
%N A135828 Expansion of psi(x^2)^8 * (psi(x)^8 + psi(-x)^8) / 2 in powers of x^2 where psi() is a Ramanujan theta function.
%C A135828 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A135828 G. C. Greubel, <a href="/A135828/b135828.txt">Table of n, a(n) for n = 0..1000</a>
%H A135828 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A135828 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A135828 Expansion of q^(-3) * ( eta(q^2)^24 + eta(q)^16 * eta(q^4)^8 ) / ( 2 * eta(q)^8 * eta(q^2)^16 / eta(q^4)^16 ) in powers of q^2.
%F A135828 7680 * a(n) = A008774(2*n + 3).
%F A135828 Convolution of A007331 and A045823.
%e A135828 G.f. = 1 + 36*x + 378*x^2 + 2200*x^3 + 8955*x^4 + 28836*x^6 + 78558*x^7 + ...
%e A135828 G.f. = q^3 + 36*q^5 + 378*q^7 + 2200*q^9 + 8955*q^11 + 28836*q^13 + 78558*q^15 + ...
%t A135828 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x]^8 (EllipticTheta[ 2, 0, x^(1/2)]^8 + EllipticTheta[ 2, Pi/4, x^(1/2)]^8 16) / 131072, {x, 0, 2 n + 3}]; (* _Michael Somos_, Oct 15 2015 *)
%o A135828 (PARI) {a(n) = my(A); if( n<0, 0, n *= 2; A = x * O(x^n); polcoeff( ( eta(x^2 + A)^24 + eta(x + A)^16 * eta(x^4 + A)^8 ) / ( 2 * eta(x + A)^8 * eta(x^2 + A)^16 / eta(x^4 + A)^16 ), n))};
%o A135828 (Magma) Basis( ModularForms( Gamma1(4), 8), 60)[4]; /* _Michael Somos_, Oct 15 2015 */
%Y A135828 Cf. A007331, A008774, A045823.
%K A135828 nonn
%O A135828 0,2
%A A135828 _Michael Somos_, Nov 29 2007