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A164617
Expansion of (phi^3(q^3) / phi(q)) * (psi(-q^3) / psi^3(-q)) in powers of q where phi(), psi() are Ramanujan theta functions.
8
1, 1, 4, 10, 20, 39, 76, 140, 244, 415, 696, 1140, 1820, 2861, 4448, 6816, 10292, 15372, 22756, 33356, 48408, 69683, 99600, 141312, 199036, 278557, 387608, 536230, 737632, 1009464, 1374888, 1863764, 2514868, 3378948, 4521672, 6027000, 8002676
OFFSET
0,3
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 eta(q^6)^14 / (eta(q) * eta(q^2)^2 * eta(q^3)^5 * eta(q^4) * eta(q^12)^5) in powers of q.
Euler transform of period 12 sequence [ 1, 3, 6, 4, 1, -6, 1, 4, 6, 3, 1, 0, ...].
Convolution of A113973 and A132974. a(n) = A164616(3*n).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
A128641(n) = (-1)^n*a(n). - Michael Somos, Apr 24 2023
EXAMPLE
G.f. = 1 + q + 4*q^2 + 10*q^3 + 20*q^4 + 39*q^5 + 76*q^6 + 140*q^7 + 244*q^8 + ...
MATHEMATICA
nmax=60; CoefficientList[Series[Product[(1-x^(6*k))^14 / ((1-x^k) * (1-x^(2*k))^2 * (1-x^(3*k))^5 * (1-x^(4*k)) * (1-x^(12*k))^5), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^14 / (eta(x + A) * eta(x^2 + A)^2 * eta(x^3 + A)^5 * eta(x^4 + A) * eta(x^12 + A)^5), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 17 2009
STATUS
approved