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A356367
Number of plane partitions of n having exactly one row and one column, each of equal length.
1
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
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
Sequence in context: A317853 A238517 A335240 * A365825 A099926 A355021
KEYWORD
nonn
AUTHOR
Jeremy Lovejoy, Oct 16 2022
STATUS
approved