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A134340 Expansion of psi(x)^3 * f(-x^3)^3 / chi(-x)^2 in powers of x where psi(), chi(), f() are Ramanujan theta functions.

%I A134340 #11 Feb 16 2025 08:33:06
%S A134340 1,5,12,22,35,50,70,92,117,145,170,210,250,287,330,362,425,477,532,
%T A134340 600,626,715,782,850,925,962,1100,1162,1247,1335,1370,1520,1617,1750,
%U A134340 1810,1850,2040,2147,2262,2380,2451,2625,2752,2882,3015,3005,3290,3500,3577
%N A134340 Expansion of psi(x)^3 * f(-x^3)^3 / chi(-x)^2 in powers of x where psi(), chi(), f() are Ramanujan theta functions.
%C A134340 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A134340 G. C. Greubel, <a href="/A134340/b134340.txt">Table of n, a(n) for n = 0..10000</a>
%H A134340 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A134340 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A134340 Expansion of q^(-5/6) * eta(q^2)^8 * eta(q^3)^3 / eta(q)^5 in powers of q.
%F A134340 Euler transform of period 6 sequence [ 5, -3, 2, -3, 5, -6, ...].
%F A134340 -24 * a(n) = A103440(6*n + 5). 216 * a(n) = A109041(6*n + 5).
%e A134340 G.f. = 1 + 5*x + 12*x^2 + 22*x^3 + 35*x^4 + 50*x^5 + 70*x^6 + 92*x^7 + 117*x^8 + ...
%e A134340 G.f. = q^5 + 5*q^11 + 12*q^17 + 22*q^23 + 35*q^29 + 50*q^35 + 70*q^41 + 94*q^47 + ...
%t A134340 a[ n_] := If[ n < 0, 0, (-1/24) DivisorSum[ 6 n + 5, #^2 KroneckerSymbol[ -3, #] &]]; (* _Michael Somos_, Oct 25 2015 *)
%t A134340 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x]^2 QPochhammer[ x^3]^3 EllipticTheta[ 2, 0, x^(1/2)]^3 / (8 x^(3/8)), {x, 0, n}]; (* _Michael Somos_, Oct 25 2015 *)
%o A134340 (PARI) {a(n) = if( n<0, 0, n = 6*n + 5; sumdiv(n, d, d^2 * kronecker( -3, d)) / -24 )};
%o A134340 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^3 + A)^3 / eta(x + A)^5, n))};
%Y A134340 Cf. A103440, A109041.
%K A134340 nonn
%O A134340 0,2
%A A134340 _Michael Somos_, Oct 21 2007