A356367 Number of plane partitions of n having exactly one row and one column, each of equal length.

1, 1, 1, 2, 2, 5, 6, 11, 16, 26, 36, 58, 81, 122, 172, 251, 350, 502, 692, 972, 1332, 1842, 2499, 3414, 4592, 6200, 8277, 11064, 14656, 19424, 25544, 33584, 43880, 57274, 74362, 96429, 124468, 160422, 205942, 263938, 337083, 429768

Offset

0,4

Comments

The empty plane partition of 0 contributes an initial term equal to 1.

Also equal to the number of unimodal compositions of n+1 where the peak appears exactly once and the number of parts to the left of the peak is equal to the number of parts to the right of the peak.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

B. Kim and J. Lovejoy, The rank of a unimodal sequence and a partial theta identity of Ramanujan, Int. J. Number Theory 10 (2014), 1081-1098.

Formula

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

Example

For n = 7 the valid unimodal compositions of n+1=8 are (8), (1,6,1), (1,5,2), (2,5,1), (3,4,1), (1,4,3), (2,4,2), (1,1,4,1,1), (1,1,3,2,1), (1,2,3,1,1) and (1,1,1,2,1,1,1), and so a(7) = 11.

Mathematica

nmax = 50; CoefficientList[Series[1 + 1/Product[(1 - x^k)^2, {k, 1, nmax}] * Sum[(-1)^(k + r + 1) * x^(k*(k + 1)/2 + r*(r + 1)/2 + 2*k*r)*(1 - x^r), {r, 0, nmax}, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 28 2023 *)

Crossrefs

Cf. A006330, A195012.

Sequence in context: A317853 A238517 A335240 * A365825 A099926 A355021

Adjacent sequences: A356364 A356365 A356366 * A356368 A356369 A356370

Keyword

nonn

Author

Jeremy Lovejoy, Oct 16 2022

Status

approved