OFFSET
1,4
COMMENTS
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Euler transform of period 30 sequence [ 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, ...].
G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)= u^2 -v -u*v*(2 + 4*v).
Expansion of (eta(q^2) * eta(q^6) * eta(q^10) * eta(q^30)) / (eta(q) * eta(q^3) * eta(q^5) * eta(q^15)) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132321.
G.f.: x * Product_{k>0} (1 + x^k) * (1 + x^(3*k)) * (1 + x^(5*k)) * (1 + x^(15*k)).
Convolution inverse of A132321.
a(n) ~ exp(2*Pi*sqrt(2*n/15)) / (2^(11/4) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
G.f.: r * s = (r + s - 1)/2 where r = x * Product_{k>0} (1 - x^(2*k)) * (1 - x^(30*k)) / ((1 - x^(3*k)) * (1 - x^(5*k))), s = Product_{k>0} (1 - x^(6*k)) * (1 - x^(10*k)) / ((1 - x^k) * (1 - x^(15*k))). - Seiichi Manyama, May 05 2017
EXAMPLE
G.f. = q + q^2 + q^3 + 3*q^4 + 3*q^5 + 5*q^6 + 8*q^7 + 9*q^8 + 13*q^9 + 19*q^10 + ...
MATHEMATICA
nmax = 40; Rest[CoefficientList[Series[x * Product[(1 + x^k) * (1 + x^(3*k)) * (1 + x^(5*k)) * (1 + x^(15*k)), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 07 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^30 + A) / (eta(x + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^15 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 03 2006, Jan 12 2009
STATUS
approved