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%I #23 Feb 16 2021 21:18:50
%S 1,4,10,26,56,126,252,512,980,1866,3427,6258,11121,19618,33975,58328,
%T 98732,165804,275246,453544,740338,1200088,1929897,3083898,4893775,
%U 7720826,12106814,18883104,29291740,45215386,69451631,106197524,161656759,245050410,369935066
%N Sum of the traces of all plane partitions of n.
%C Convolution of A000203 and A000219. - _Vaclav Kotesovec_, Sep 25 2016
%C Convolution of A340793 and A091360. - _Omar E. Pol_, Feb 16 2021
%D G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, pp. 179-201.
%H Vaclav Kotesovec, <a href="/A276432/b276432.txt">Table of n, a(n) for n = 1..10000</a>
%F G.f.: g(x) = Sum_{j>=1} (j*x^j/(1-x^j))/Product_{k>=1} (1-x^k)^k.
%F a(n) = Sum(k*A089353(n,k), k>=1).
%e a(3) = 10 because the 6 (=A000219(3)) planar partitions of 3 are [3], [2,1], [2;1], [1,1,1], [1;1;1], [1,1;1] (; indicates a new row); the sum of their traces is 3+2+2+1+1+1 = 10.
%p g:= (sum(j*x^j/(1-x^j),j = 1..100))/(product((1-x^k)^k,k = 1..100)): gser := series(g, x = 0,40): seq(coeff(gser, x, m), m = 1 .. 35);
%p # second Maple program:
%p b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0, add((p
%p ->p+[0, j*p[1]])(b(n-i*j, i-1))*binomial(i+j-1, j), j=0..n/i)))
%p end:
%p a:= n-> b(n$2)[2]:
%p seq(a(n), n=1..50); # _Alois P. Heinz_, Sep 24 2018
%t nmax = 50; Rest[CoefficientList[Series[Sum[j*x^j/(1-x^j), {j, 1, nmax}]*Product[1/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Sep 25 2016 *)
%Y Cf. A000219, A089353, A277029.
%Y Cf. A091360, A340793.
%K nonn
%O 1,2
%A _Emeric Deutsch_, Sep 24 2016
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