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A326827
Expansion of 1 / (chi(-x)^10 * chi(-x^2)^4) in powers of x where chi() is a Ramanujan theta function.
0
1, 10, 59, 270, 1045, 3582, 11194, 32488, 88716, 230150, 571363, 1365148, 3153522, 7069242, 15425719, 32849906, 68421073, 139645914, 279740407, 550790788, 1067244261, 2037348726, 3835457084, 7126887974, 13081454919, 23735283778, 42598577587, 75668099822
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^(-3/4) * (eta(q^2)^3 * eta(q^4)^2 / eta(q)^5)^2 in powers of q.
Euler transform of period 4 sequence [10, 4, 10, 0, ...].
G.f.: Product_{n>=0} (1 - x^(2*n + 1))^-10 * (1 - x^(4*n + 2))^-4.
A093160(2*n + 1) = A123655(4*n + 3) = 4*a(n).
A232772(2*n + 1) = A215348(4*n + 3) = A215349(4*n + 3) = 8*a(n).
A007096(4*n + 3) = A212318(4*n + 3) = 16*a(n). A189925(4*n + 3) = A232358(4*n + 3) = -16*a(n).
a(n) ~ exp(2*Pi*sqrt(n)) / (256*n^(3/4)). - Vaclav Kotesovec, Oct 31 2019
EXAMPLE
G.f. = 1 + 10*x + 59*x^2 + 270*x^3 + 1045*x^4 + 3582*x^5 + 11194*x^6 + ...
G.f. = q^3 + 10*q^7 + 59*q^11 + 270*q^15 + 1045*q^19 + 3582*q^23 + 11194*q^27 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^3 QPochhammer[ x^4]^2 / (QPochhammer[ x]^5))^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ x^(-3/4) (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x] / EllipticTheta[ 4, 0, x]^2 / 4)^2, {x, 0, n}];
nmax = 20; CoefficientList[Series[Product[(1 + x^k)^10/(1 - x^(4*k - 2))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 31 2019 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 * eta(x^4 + A)^2 / eta(x + A)^5)^2, n))};
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 20 2019
STATUS
approved