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

0,3

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

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S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 10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

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

Example

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 + ...

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 *)

Prog

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

Crossrefs

Convolution with A002448 is A132211.

Cf. A015128.

Sequence in context: A342091 A239288 A306145 * A309173 A116084 A108925

Adjacent sequences: A143181 A143182 A143183 * A143185 A143186 A143187

Keyword

nonn

Author

Michael Somos, Jul 28 2008

Status

approved