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

0,3

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 370, 9th equation.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

Cf. A015128, A059777.

Sequence in context: A018586 A135342 A029877 * A147425 A080430 A053523

Adjacent sequences: A207638 A207639 A207640 * A207642 A207643 A207644

Keyword

nonn

Author

Paul D. Hanna, Feb 19 2012

Status

approved