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%I #11 Apr 28 2023 07:01:14
%S 1,1,1,2,2,5,6,11,16,26,36,58,81,122,172,251,350,502,692,972,1332,
%T 1842,2499,3414,4592,6200,8277,11064,14656,19424,25544,33584,43880,
%U 57274,74362,96429,124468,160422,205942,263938,337083,429768
%N Number of plane partitions of n having exactly one row and one column, each of equal length.
%C The empty plane partition of 0 contributes an initial term equal to 1.
%C 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.
%H Vaclav Kotesovec, <a href="/A356367/b356367.txt">Table of n, a(n) for n = 0..1000</a>
%H B. Kim and J. Lovejoy, <a href="https://doi.org/10.1142/S179304211450016X">The rank of a unimodal sequence and a partial theta identity of Ramanujan</a>, Int. J. Number Theory 10 (2014), 1081-1098.
%F 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).
%e 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.
%t 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 *)
%Y Cf. A006330, A195012.
%K nonn
%O 0,4
%A _Jeremy Lovejoy_, Oct 16 2022
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