OFFSET
0,3
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
In Ramanujan's equation let a = x and b = 1. - Michael Somos, Nov 20 2015
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 370, 9th equation.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..750 from Vaclav Kotesovec)
Frank Garvan, Dyson's rank function and Andrews's SPT-function, slides 29, 30.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of 1 / ((1 + x) * phi(-x)) in powers of x where phi() is a Ramanujan theta function. - Michael Somos, Nov 20 2015
G.f.: 1 + x*(1+x) * (1 / (1-x)^2 + 2*x^3 / ((1-x)*(1-x^2))^2 + 2*x^7*(1+x) / ((1-x)*(1-x^2)*(1-x^3))^2 + 2*x^12*(1+x)*(1+x^2) / ((1-x)*(1-x^2)*(1-x^3)*(1-x^4))^2 + ...). [Ramanujan] - Michael Somos, Nov 20 2015
a(n) + a(n+1) = A015128(n+1) for n >= 0. - Seiichi Manyama, Jul 12 2018
a(n) ~ exp(Pi*sqrt(n)) / (16*n). - Vaclav Kotesovec, Jun 18 2019
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 5*x^3 + 9*x^4 + 15*x^5 + 25*x^6 + 39*x^7 +...
such that, by definition,
A(x) = 1 + x*(1+x)/(1-x) + x^2*(1+x)*(1+x^2)/((1-x)*(1-x^2)) + x^3*(1+x)*(1+x^2)*(1+x^3)/((1-x)*(1-x^2)*(1-x^3)) +...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {-x}, {}, x, x], {x, 0, n}]; (* Michael Somos, Mar 11 2014 *)
a[ n_] := SeriesCoefficient[ 1 / ((1 + x) EllipticTheta[ 4, 0, x]), {x, 0, n}]; (* Michael Somos, Nov 20 2015 *)
PROG
(PARI) {a(n)=polcoeff(sum(m=0, n, x^m*prod(k=1, m, (1+x^k)/(1-x^k +x*O(x^n))) ), n)}
for(n=0, 50, print1(a(n), ", "))
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / ((1 + x) * eta(x + A)^2), n))}; /* Michael Somos, Nov 20 2015 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 19 2012
STATUS
approved