OFFSET
-1,3
LINKS
G. C. Greubel, Table of n, a(n) for n = -1..1000
FORMULA
Expansion of (eta(q^3) * eta(q^13)) / (eta(q) * eta(q^39)) in powers of q.
Euler transform of period 39 sequence [ 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) = f(1/A(x), 1/A(x^2)) where f(u, v) = u^3 + v^3 + 2*u*v*(u + v) - u^2*v^2 - u*v.
G.f.: x^-1 * Product_{k>0} (1 - x^(3*k)) * (1 - x^(13*k)) / ((1 - x^k) * (1 - x^(39*k))).
a(n) ~ exp(4*Pi*sqrt(n/39)) / (sqrt(2) * 39^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015
EXAMPLE
1/q + 1 + 2*q + 2*q^2 + 4*q^3 + 5*q^4 + 7*q^5 + 9*q^6 + 13*q^7 + 16*q^8 + 22*q^9 + ...
MATHEMATICA
QP = QPochhammer; s = QP[q^3]*(QP[q^13]/(QP[q]*QP[q^39])) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from PARI *)
PROG
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^13 + A) / (eta(x + A) * eta(x^39 + A)), n))}
(PARI) {a(n) = local(A, u, v); if( n<-1, 0, A = 1/x; for( k=0, n, u = A + x * O(x^k); v = subst(u, x, x^2); A += x^k * polcoeff( u^3 + v^3 + 2*u*v*(u + v) - u^2*v^2 - u*v, k-5) / 2); polcoeff(A, n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, May 03 2004
STATUS
approved