cp's OEIS Frontend

A143184 Coefficients of a Ramanujan q-series.

%I A143184 #29 May 13 2025 22:45:42
%S A143184 1,1,2,4,6,10,15,23,33,49,69,98,136,188,256,348,466,622,824,1084,1418,
%T A143184 1846,2389,3077,3947,5038,6407,8115,10241,12876,16141,20160,25110,
%U A143184 31179,38609,47674,58724,72141,88421,108114,131902,160565,195061,236468
%N A143184 Coefficients of a Ramanujan q-series.
%C A143184 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
%C A143184 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
%D A143184 S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 10
%H A143184 Alois P. Heinz, <a href="/A143184/b143184.txt">Table of n, a(n) for n = 0..10000</a>
%H A143184 G. E. Andrews, <a href="https://doi.org/10.1016/0001-8708(84)90017-3">Ramanujan's "lost" notebook. IV. Stacks and alternating parity in partitions</a>, Adv. in Math. 53 (1984), 55-74.
%H A143184 B. Kim, E. Kim, and J. Lovejoy, <a href="https://doi.org/10.1142/S1793042120400345">On weighted overpartitions related to some q-series in Ramanujan's lost notebook</a>, Int. J. Number Theory 17 (2021), 603-619.
%F A143184 G.f.: Sum_{k>=0} x^((k^2+k)/2) / ((1 - x) * (1 - x^2) ... (1 - x^k))^2.
%F A143184 a(n) ~ exp(2*Pi*sqrt(n/5)) / (2^(3/2) * 5^(3/4) * n). - _Vaclav Kotesovec_, Nov 20 2020
%e A143184 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 + ...
%p A143184 b:= proc(n, i) option remember;
%p A143184      `if`(i>n, 0, `if`(irem(n, i, 'r')=0, r, 0)+
%p A143184       add(j*b(n-i*j, i+1), j=1..n/i))
%p A143184     end:
%p A143184 a:= n-> `if`(n=0, 1, b(n, 1)):
%p A143184 seq(a(n), n=0..50);  # _Alois P. Heinz_, Oct 03 2018
%t A143184 m = 50;
%t A143184 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 *)
%o A143184 (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))};
%Y A143184 Convolution with A002448 is A132211.
%Y A143184 Cf. A015128 (has definition of overpartitions).
%K A143184 nonn
%O A143184 0,3
%A A143184 _Michael Somos_, Jul 28 2008