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A259910
Expansion of f(-x^2, -x^3)^3 / f(-x)^2 in powers of x where f(,) is the Ramanujan general theta function.
4
1, 2, 2, 1, 2, 3, 4, 3, 3, 4, 6, 6, 7, 8, 8, 9, 11, 12, 13, 14, 17, 19, 21, 21, 25, 27, 30, 31, 35, 39, 43, 47, 51, 55, 60, 65, 71, 77, 83, 88, 98, 105, 115, 122, 132, 142, 155, 164, 178, 191, 206, 220, 236, 252, 272, 290, 311, 332, 356, 378, 407, 434, 464
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 6th equation.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)
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) = 3 * A053266(n) - A053262(n) unless n=0.
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
Sequence in context: A352888 A354267 A366804 * A072549 A239481 A200114
KEYWORD
nonn
AUTHOR
Michael Somos, Jul 07 2015
STATUS
approved