A143184 Coefficients of a Ramanujan q-series.

1, 1, 2, 4, 6, 10, 15, 23, 33, 49, 69, 98, 136, 188, 256, 348, 466, 622, 824, 1084, 1418, 1846, 2389, 3077, 3947, 5038, 6407, 8115, 10241, 12876, 16141, 20160, 25110, 31179, 38609, 47674, 58724, 72141, 88421, 108114, 131902, 160565, 195061, 236468

Offset: 0

Author

Michael Somos, Jul 28 2008

Keywords

• nonn

Comments

Also equal to the number of overpartitions of n with no non-overlined parts larger than the number of overlined parts. For example, the overpartitions counted by a(4) = 6 are: [4'], [3',1], [3',1'], [2',1,1], [2',1',1], [1',1,1,1]. - Jeremy Lovejoy, Aug 23 2021

Also the number of partitions into red and blue integers, where the red parts cover an initial segment of the positive integers, and there can be no blue integer if there is not also a red one with the same value. For example, when the red integers are marked with a prime, the partitions counted by a(4) are: [2',1',1], [2',1',1'], [1',1,1,1], [1',1',1,1], [1',1',1',1], [1',1',1',1']. - Christian Sievers, May 09 2025

Examples

			G.f. = 1 + q + 2*q^2 + 4*q^3 + 6*q^4 + 10*q^5 + 15*q^6 + 23*q^7 + 33*q^8 + ...
		

References

• S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 10

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• G. E. Andrews, Ramanujan's "lost" notebook. IV. Stacks and alternating parity in partitions, Adv. in Math. 53 (1984), 55-74.
• B. Kim, E. Kim, and J. Lovejoy, On weighted overpartitions related to some q-series in Ramanujan's lost notebook, Int. J. Number Theory 17 (2021), 603-619.

Crossrefs

Convolution with A002448 is A132211.

Cf. A015128 (has definition of overpartitions).

Programs

Maple

b:= proc(n, i) option remember;
     `if`(i>n, 0, `if`(irem(n, i, 'r')=0, r, 0)+
      add(j*b(n-i*j, i+1), j=1..n/i))
    end:
a:= n-> `if`(n=0, 1, b(n, 1)):
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 03 2018

Mathematica

m = 50;
Sum[x^(k(k+1)/2)/Product[1-x^j, {j, 1, k}]^2, {k, 0, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 20 2020 *)

PARI

{a(n) = my(t); if(n<0, 0, t = 1 + x*O(x^n); polcoef(sum(k=1, (sqrtint(8*n + 1) - 1)\2, t = t*x^k/(1 - x^k)^2 + x*O(x^n), 1), n))};

Formula

G.f.: Sum_{k>=0} x^((k^2+k)/2) / ((1 - x) * (1 - x^2) ... (1 - x^k))^2.

a(n) ~ exp(2*Pi*sqrt(n/5)) / (2^(3/2) * 5^(3/4) * n). - Vaclav Kotesovec, Nov 20 2020