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A113261 Expansion of (9*phi(q)*phi(q^3)^5 - phi(q)^5*phi(q^3))/8 in powers of q where phi(q) is a Ramanujan theta function.

%I A113261 #22 Feb 16 2025 08:32:59
%S A113261 1,1,-5,1,11,-24,-5,50,-53,1,120,-120,11,170,-250,-24,203,-288,-5,362,
%T A113261 -264,50,600,-528,-53,601,-850,1,550,-840,120,962,-821,-120,1440,
%U A113261 -1200,11,1370,-1810,170,1272,-1680,-250,1850,-1320,-24,2640,-2208,203,2451,-3005,-288,1870,-2808
%N A113261 Expansion of (9*phi(q)*phi(q^3)^5 - phi(q)^5*phi(q^3))/8 in powers of q where phi(q) is a Ramanujan theta function.
%C A113261 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A113261 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 226 Entry 4(ii).
%H A113261 G. C. Greubel, <a href="/A113261/b113261.txt">Table of n, a(n) for n = 0..1000</a>
%H A113261 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A113261 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A113261 Euler transform of period 12 sequence [1, -6, 6, -5, 1, -12, 1, -5, 6, -6, 1, -6, ...].
%F A113261 Expansion of eta(q^2)^7 * eta(q^6)^11 / (eta(q) * eta(q^3)^5 * eta(q^4) * eta(q^12)^5) in powers of q.
%F A113261 a(n) is multiplicative and a(3^e) = 1, a(2^e) = ((-4)^(e+1)-1)/(-4-1)-2, a(p^e) = ((-p^2)^(e+1)-1)/(-p-1) if p == 2 (mod 3), a(p^e) = ((p^2)(e+1)-1)/(p-1) if p == 1 (mod 3).
%F A113261 G.f.: 1 + Sum_{k>0} k^2 x^k/(1-(-x)^k) Kronecker(-3, k).
%e A113261 G.f. = 1 + q - 5*q^2 + q^3 + 11*q^4 - 24*q^5 - 5*q^6 + 50*q^7 + ... - _Michael Somos_, Sep 07 2018
%t A113261 a[n_]:= SeriesCoefficient[(9*EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^3]^5 - EllipticTheta[3, 0, q]^5*EllipticTheta[3, 0, q^3])/8, {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Feb 09 2018 *)
%o A113261 (PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, d^2 * kronecker(-3, d) * (-1)^(n-d)))};
%o A113261 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^6 + A)^11 / (eta(x + A) * eta(x^3 + A)^5 * eta(x^4 + A) * eta(x^12 + A)^5), n))};
%o A113261 (PARI) {a(n) = my(A, p, e, t); if( n<1, n==0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 1, t=p^2*(-1)^(p%3==2); (t^(e+1) - 1) / (t-1) - 2*(p==2))))};
%K A113261 sign,mult
%O A113261 0,3
%A A113261 _Michael Somos_, Oct 21 2005