A207641 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1+x^k)/(1-x^k).

1, 1, 3, 5, 9, 15, 25, 39, 61, 93, 139, 205, 299, 429, 611, 861, 1201, 1663, 2285, 3115, 4221, 5683, 7605, 10123, 13405, 17661, 23163, 30245, 39323, 50925, 65699, 84445, 108167, 138089, 175719, 222921, 281965, 355627, 447309, 561139, 702133, 876395, 1091301

Offset: 0

Author

Paul D. Hanna, Feb 19 2012

Keywords

• nonn

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

Examples

			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)) +...
		

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• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 370, 9th equation.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

Crossrefs

Cf. A015128, A059777.

Programs

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 *)

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 */

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