OFFSET
0,2
COMMENTS
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 6th equation.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of f(-x) * (f(-x^5) / f(-x, -x^4))^3 in powers of x where f(,) is the Ramanujan general theta function.
Expansion of f(-x^2, -x^3) * G(x)^2 in powers of x where f(,) is the Ramanujan general theta function and G() is a Rogers-Ramanujan function. - Michael Somos, Jul 09 2015
Euler transform of period 5 sequence [ 2, -1, -1, 2, -1, ...].
G.f.: (Sum_{k in Z} (-1)^k * x^(k*(5*k + 1)/2)) / (Product_{k in Z} 1 - x^abs(5*k + 1))^2.
a(n) ~ sqrt(5+2*sqrt(5)) * exp(sqrt(2*n/15)*Pi)/ (5*sqrt(2*n)). - Vaclav Kotesovec, Dec 17 2016
EXAMPLE
G.f. = 1 + 2*x + 2*x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 3*x^7 + 3*x^8 + ...
G.f. = 1/q + 2*q^119 + 2*q^239 + q^359 + 2*q^479 + 3*q^599 + 4*q^719 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ -2, 1, 1, -2, 1}[[
Mod[k, 5, 1]]], {k, n}], {x, 0, n}];
nmax = 100; CoefficientList[Series[Product[(1-x^k)/((1 - x^(5*k-1))*(1 - x^(5*k-4)))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 17 2016 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, -2, 1, 1, -2][k%5+1]), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jul 07 2015
STATUS
approved