A235790 -id:A235790 - cp's OEIS frontend

A335666 a(n) is the sum, over all overpartitions of n, of the overlined parts.

1, 3, 10, 21, 46, 90, 168, 295, 511, 850, 1382, 2198, 3430, 5260, 7960, 11861, 17468, 25445, 36670, 52346, 74092, 103986, 144840, 200322, 275191, 375662, 509816, 687960, 923442, 1233340, 1639312, 2168999, 2857460, 3748772, 4898652, 6377023, 8271294, 10690830, 13771912, 17683642

Offset: 1

Jeremy Lovejoy, Jun 17 2020

nonn

			The 8 overpartitions of 8 are [3], [3'], [2,1], [2,1'], [2',1], [2',1'], [1,1,1], [1',1,1], and so a(3) = 10.

K. Bringmann, J. Lovejoy, and R. Osburn, Rank and crank moments for overpartitions, Journal of Number Theory, 129 (2009), 1758-1772.

Cf. A015128, A230441, A235790, A235792, A235793, A235798, A236000, A236001, A237047, A335651.

Cf. A305101 (number of overlined parts).

my(N=44, q='q+O('q^N)); Vec( prod(k=1,N, (1+q^k)/(1-q^k)) * sum(k=1,N, k*q^k/(1+q^k)) ) \\ Joerg Arndt, Jun 18 2020

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

a(n) = A235793(n) - A335651(n). - Omar E. Pol, Jun 17 2020

A359481 Irregular triangle read by rows in which T(n,k) is one half of the number of overpartitions of n having k distinct parts, n>=1, k>=1.

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 2, 10, 4, 12, 4, 2, 22, 8, 4, 26, 20, 3, 34, 40, 4, 44, 60, 8, 2, 54, 100, 16, 6, 58, 148, 40, 2, 74, 208, 80, 4, 88, 268, 160, 4, 88, 388, 240, 16, 5, 110, 468, 416, 32, 2, 118, 616, 616, 80, 6, 136, 736, 936, 160, 2, 142, 940, 1296, 320, 6, 162, 1108, 1816, 576

Offset: 1

Omar E. Pol, Mar 31 2023

			Triangle begins:
  1;
  2;
  2,  2;
  3,  4;
  2, 10;
  4, 12,   4;
  2, 22,   8;
  4, 26,  20;
  3, 34,  40;
  4, 44,  60,   8;
  2, 54, 100,  16;
  6, 58, 148,  40;
  2, 74, 208,  80;
  4, 88, 268, 160;
  4, 88, 388, 240, 16;
  ...

Column 1 gives A000005.
Row n has length A003056(n).
Row sums give A014968, n >= 1.
The first element of column k is A000079(k-1).
The first element of column k is in row A000217(k).
Cf. A116608, A235790.

T(n,k) = (1/2)*A235790(n,k).

T(n,k) = (2^(k-1))*A116608(n,k).

cp's OEIS Frontend

A335666 a(n) is the sum, over all overpartitions of n, of the overlined parts.

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