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A169976 Expansion of (psi(x)^24 + psi(-x)^24) / 2 in powers of x^2 where psi() is a Ramanujan theta function.

%I A169976 #10 Jan 11 2025 03:09:35
%S A169976 1,276,11178,177400,1612875,10131156,48897678,193740408,658523925,
%T A169976 1980143600,5386270686,13477895856,31425764410,68969957700,
%U A169976 143635113000,285718115112,545796171084,1005775268868,1794713445350,3111031518000
%N A169976 Expansion of (psi(x)^24 + psi(-x)^24) / 2 in powers of x^2 where psi() is a Ramanujan theta function.
%H A169976 G. C. Greubel, <a href="/A169976/b169976.txt">Table of n, a(n) for n = 0..1000</a>
%F A169976 a(n) = (A013959(2*n + 3) - A000594(2*n + 3)) / 176896 = A014809(2*n).
%e A169976 1 + 276*x + 11178*x^2 + 177400*x^3 + 1612875*x^4 + 10131156*x^5 + ...
%e A169976 q^3 + 276*q^5 + 11178*q^7 + 177400*q^9 + 1612875*q^11 + 10131156*q^13 + ...
%t A169976 QP:= Pochhammer; a[n_]:= SeriesCoefficient[(QP[q, q])^24*(QP[-q^(1/2), q^(1/2)]^24 + QP[q^(1/2), -q^(1/2)]^24)/2, {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Apr 04 2018 *)
%t A169976 a[n_] := (DivisorSigma[11, 2*n+3] - RamanujanTau[2*n+3]) / 176896; Array[a, 20, 0] (* _Amiram Eldar_, Jan 11 2025 *)
%o A169976 (PARI) {a(n) = local(A); if( n<0, 0, n *= 2; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A))^24, n))}
%o A169976 (PARI) {a(n) = if( n<0, 0, n = 2*n + 3; (sigma(n, 11) - polcoeff( x * eta(x + x * O(x^n))^24, n)) / 176896 )}
%Y A169976 Cf. A000594, A013959, A014809.
%K A169976 nonn
%O A169976 0,2
%A A169976 _Michael Somos_, Aug 15 2010