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A259033
Expansion of psi(x^3)^2 * f(-x^2)^4 / f(-x)^6 in powers of where psi(), f() are Ramanujan theta function.
2
1, 6, 23, 76, 221, 584, 1443, 3368, 7497, 16046, 33190, 66628, 130288, 248858, 465387, 853836, 1539425, 2731462, 4775703, 8236856, 14027754, 23609794, 39301171, 64747876, 105638153, 170778512, 273704800, 435079524, 686237877, 1074405242, 1670333294, 2579418528
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-5/6) * (eta(q^2)^2 * eta(q^6)^2 / (eta(q)^3 * eta(q^3)))^2 in powers of q.
Euler transform of period 6 sequence [ 6, 2, 8, 2, 6, 0, ...].
a(n) = A263528(3*n + 2). -2 * a(n) = A261369(2*n + 1) = A213265(6*n + 5) = A262930(6*n + 5).
a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(19/4) * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017
EXAMPLE
G.f. = 1 + 6*x + 23*x^2 + 76*x^3 + 221*x^4 + 584*x^5 + 1443*x^6 + 3368*x^7 + ...
G.f. = q^5 + 6*q^11 + 23*q^17 + 76*q^23 + 221*q^29 + 584*q^35 + 1443*q^41 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (1/4) x^(-3/4) EllipticTheta[ 2, 0, x^(3/2)]^2 QPochhammer[ x^2]^4 / QPochhammer[ x]^6, {x, 0, n}];
nmax = 40; CoefficientList[Series[Product[((1 + x^k)^2 * (1 + x^(3*k))^2 * (1 - x^(3*k)) / (1 - x^k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 16 2017 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^6 + A)^2 / (eta(x + A)^3 * eta(x^3 + A)))^2, n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Nov 07 2015
STATUS
approved