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Expansion of psi(x^2) * psi(x^3) in powers of x where psi() is a Ramanujan theta function.
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%I #14 Feb 16 2025 08:33:26

%S 1,0,1,1,0,1,1,0,0,2,0,1,1,0,0,2,0,0,1,0,2,1,0,1,1,0,0,0,0,1,3,0,1,1,

%T 0,0,1,0,1,1,0,0,2,0,0,2,0,1,1,0,1,2,0,0,0,0,1,1,0,1,2,0,0,1,0,3,0,0,

%U 0,1,0,0,2,0,1,3,0,0,0,0,0,1,0,1,1,0,2

%N Expansion of psi(x^2) * psi(x^3) in powers of x where psi() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A259895/b259895.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of q^(-5/8) * eta(q^4)^2 * eta(q^6)^2 / (eta(q^2) * eta(q^3)) in powers of q.

%F Euler transform of period 12 sequence [ 0, 1, 1, -1, 0, 0, 0, -1, 1, 1, 0, -2, ...].

%F a(n) = A259896(3*n + 1). a(3*n) = A128583(n). a(3*n + 1) = a(9*n + 8) = 0.

%F 2 * a(n) = A129402(4*n + 2) = A190615(4*n + 2) = A000377(8*n + 5) = A192013(8*n + 5). - _Michael Somos_, Jul 22 2015

%F -2 * a(n) = A259668(2*n + 1) = A128580(4*n + 2) = A134177(4*n + 2) = A257921(6*n + 3). - _Michael Somos_, Jul 22 2015

%F a(3*n + 2) = A259896(n). - _Michael Somos_, Jul 22 2015

%e G.f. = 1 + x^2 + x^3 + x^5 + x^6 + 2*x^9 + x^11 + x^12 + 2*x^15 + x^18 + ...

%e G.f. = q^5 + q^21 + q^29 + q^45 + q^53 + 2*q^77 + q^93 + q^101 + 2*q^125 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^(3/2)] / (4 q^(5/8)), {x, 0, n}];

%t a[ n_] := If[ n < 0, 0, 1/2 Sum[ KroneckerSymbol[ -6, d], {d, Divisors[8 n + 5]}]]; (* _Michael Somos_, Jul 22 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A)), n))};

%o (PARI) {a(n) = if( n<0, 0, 1/2 * sumdiv( 8*n + 5, d, kronecker( -6, d)))};

%Y Cf. A000377, A128580, A128583, A129402, A134177, A190615, A192013, A257921, A259668, A259896.

%K nonn,changed

%O 0,10

%A _Michael Somos_, Jul 07 2015