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A213791 Expansion of psi(-x)^6 in powers of x where psi() is a Ramanujan theta function.

%I A213791 #18 Feb 16 2025 08:33:17
%S A213791 1,-6,15,-26,45,-66,82,-120,156,-170,231,-276,290,-390,435,-438,561,
%T A213791 -630,651,-780,861,-842,1020,-1170,1095,-1326,1431,-1370,1716,-1740,
%U A213791 1682,-2016,2145,-2132,2415,-2550,2353,-2850,3120,-2810,3321,-3486,3285,-3906,4005
%N A213791 Expansion of psi(-x)^6 in powers of x where psi() is a Ramanujan theta function.
%C A213791 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A213791 J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), pp. 111-112. see Eq. X
%D A213791 J. W. L. Glaisher, Notes on Certain Formulae in Jacobi's Fundamenta Nova, Messenger of Mathematics, 5 (1876), pp. 174-179. see p.176
%H A213791 G. C. Greubel, <a href="/A213791/b213791.txt">Table of n, a(n) for n = 0..1000</a>
%H A213791 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A213791 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A213791 Expansion of q^(-3/4) * ( eta(q) * eta(q^4) / eta(q^2) )^6 in powers of q.
%F A213791 Expansion of -1/(8 * r) * ( 1^2 * r^1 / (1 + q) - 3^2 * q^(3/4) / (1 + q^3) - 5^2 * r^5 / (1 + q^5) + 7^2 * q^(7/4) / (1 + q^7) + 9^2 * r^9 / (1 + q^9) - ...) in powers of q where r = q^(3/4) [Glaisher 1876].
%F A213791 Expansion of q^(-1/4) * ( sqrt(k * k') * K / Pi )^3 in powers of q where k, k', K are Jacobi elliptic functions. [Jacobi 1828, p. 108 quoted in Glaisher 1876, p. 176].
%F A213791 Euler transform of period 4 sequence [ -6, 0, -6, -6, ...].
%F A213791 G.f.: (Sum_{k>0} (-x)^((k^2 - k)/2))^6.
%F A213791 G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 64^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
%F A213791 a(n) = (-1)^n * A008440(n). Convolution cube of A134343.
%e A213791 G.f. = 1 - 6*x + 15*x^2 - 26*x^3 + 45*x^4 - 66*x^5 + 82*x^6 - 120*x^7 + ...
%e A213791 G.f. = q^3 - 6*q^7 + 15*q^11 - 26*q^15 + 45*q^19 - 66*q^23 + 82*q^27 + ...
%t A213791 a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[ x^2, x^4])^6, {x, 0, n}]; (* _Michael Somos_, Jun 10 2015 *)
%o A213791 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A) * eta(x^4 + A) / eta(x^2 + A) )^6, n))};
%Y A213791 Cf. A008440, A134343.
%K A213791 sign
%O A213791 0,2
%A A213791 _Michael Somos_, Jun 20 2012