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A164612
Expansion of q^(-1) * phi^2(q) * chi^3(q^9) / (chi(q^3) * phi^2(q^9)) in powers of q where phi(), chi() are Ramanujan theta functions.
4
1, 4, 4, -1, 0, 4, 1, 0, 0, 1, 0, -8, -1, 0, -8, 0, 0, 4, 1, 0, 16, -2, 0, 16, 0, 0, -4, 2, 0, -32, -3, 0, -32, 1, 0, 8, 4, 0, 56, -4, 0, 56, 1, 0, -16, 4, 0, -96, -6, 0, -92, 1, 0, 24, 5, 0, 160, -8, 0, 152, 1, 0, -40, 8, 0, -252, -10, 0, -240, 2, 0, 64, 11, 0, 392, -14, 0, 368, 4, 0, -96, 14, 0, -600, -19, 0, -560, 4, 0
OFFSET
-1,2
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 chi(q^3)^3 / (q * chi(q)) + 4 + 4 * q * chi(q) / chi(q^3)^3 in powers of q where chi() is a Ramanujan theta function.
Expansion of eta(q^2)^10 * eta(q^3) * eta(q^9) * eta(q^12) * eta(q^36) / (eta(q)^4 * eta(q^4)^4 * eta(q^6)^2 * eta(q^18)^4) in powers of q.
Euler transform of period 36 sequence [ 4, -6, 3, -2, 4, -5, 4, -2, 2, -6, 4, -2, 4, -6, 3, -2, 4, -2, 4, -2, 3, -6, 4, -2, 4, -6, 2, -2, 4, -5, 4, -2, 3, -6, 4, 0, ...].
a(3*n) = 0 unless n=0. a(n) = A164268(n) unless n=0.
Convolution of A164613 and A062244.
EXAMPLE
G.f. = 1/q + 4 + 4*q - q^2 + 4*q^4 + q^5 + q^8 - 8*q^10 - q^11 - 8*q^13 + ...
MATHEMATICA
eta[x_] := QPochhammer[x]; A164612[n_] := SeriesCoefficient[eta[q^2]^10* eta[q^3]*eta[q^9]*eta[q^12]*eta[q^36]/(eta[q]^4*eta[q^4]^4*eta[q^6]^2 *eta[q^18]^4), {q, 0, n}]; Table[A164612[n], {n, 0, 50}] (* G. C. Greubel, Aug 10 2017 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^10 * eta(x^3 + A) * eta(x^9 + A) * eta(x^12 + A) * eta(x^36 + A) / (eta(x + A)^4 * eta(x^4 + A)^4 * eta(x^6 + A)^2 * eta(x^18 + A)^4), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Aug 17 2009
STATUS
approved