OFFSET
1,2
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
Expansion of (eta(q^2) * eta(q^10) / (eta(q) * eta(q^5)))^4 in powers of q.
Euler transform of period 10 sequence [ 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v*(1 + 8*u + 16*u*v).
G.f.: x * (Product_{k>0} (1 - x^(10*k - 5)) * (1 - x^(2*k - 1)))^-4.
Convolution inverse of A132040. - Michael Somos, Apr 26 2015
a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (16 * 2^(3/4) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
EXAMPLE
G.f. = q + 4*q^2 + 10*q^3 + 24*q^4 + 51*q^5 + 104*q^6 + 206*q^7 + 384*q^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ q (QPochhammer[ -q, q] QPochhammer[ -q^5, q^5] )^4, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
nmax = 40; Rest[CoefficientList[Series[x * Product[1/((1 - x^(10*k - 5)) * (1 - x^(2*k - 1)))^4, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 07 2015 *)
PROG
(PARI) {a(n) = if( n<1, 0, n--; polcoeff( (1 / prod(k=1, (n+5)\10, 1 - x^(10*k - 5), 1 + x * O(x^n)) / prod(k=1, (n+1)\2, 1 - x^(2*k - 1), 1 + x * O(x^n)))^4, n))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^10 + A) / (eta(x + A) * eta(x^5 + A)))^4, n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Apr 17 2004, Oct 04 2004
EXTENSIONS
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 05 2007
STATUS
approved