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Triangular array read by rows: T(n,k) is the number of weakly unimodal partitions of n in which the greatest part occurs exactly k times, n>=1, 1<=k<=n.
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%I #31 Oct 24 2018 16:34:35

%S 1,1,1,3,0,1,6,1,0,1,12,2,0,0,1,21,4,1,0,0,1,38,6,2,0,0,0,1,63,11,3,1,

%T 0,0,0,1,106,16,5,2,0,0,0,0,1,170,27,7,3,1,0,0,0,0,1,272,40,11,4,2,0,

%U 0,0,0,0,1,422,63,16,6,3,1,0,0,0,0,0,1,653,92,24,8,4,2,0,0,0,0,0,0,1,986,141,34,12,5,3,1,0,0,0,0,0,0,1

%N Triangular array read by rows: T(n,k) is the number of weakly unimodal partitions of n in which the greatest part occurs exactly k times, n>=1, 1<=k<=n.

%C These are called stack polyominoes in the Flajolet and Sedgewick reference.

%D P. Flajolet and R Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 46.

%H Alois P. Heinz, <a href="/A247255/b247255.txt">Rows n = 1..141, flattened</a>

%F G.f.: Sum_{k>=1} y*x^k/(1 - y*x^k)/(Product_{i=1..k-1} (1 - x^i))^2.

%F For fixed k>=1, T(n,k) ~ Pi^(k-1) * (k-1)! * exp(2*Pi*sqrt(n/3)) / (2^(k+2) * 3^(k/2 + 1/4) * n^(k/2 + 3/4)). - _Vaclav Kotesovec_, Oct 24 2018

%e 1;

%e 1, 1;

%e 3, 0, 1;

%e 6, 1, 0, 1;

%e 12, 2, 0, 0, 1;

%e 21, 4, 1, 0, 0, 1;

%e 38, 6, 2, 0, 0, 0, 1;

%e 63, 11, 3, 1, 0, 0, 0, 1;

%e 106, 16, 5, 2, 0, 0, 0, 0, 1;

%e 170, 27, 7, 3, 1, 0, 0, 0, 0, 1;

%p b:= proc(n, i) option remember; local r; expand(

%p `if`(i>n, 0, `if`(irem(n, i, 'r')=0, x^r, 0)+

%p add(b(n-i*j, i+1)*(j+1), j=0..n/i)))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 1)):

%p seq(T(n), n=1..14); # _Alois P. Heinz_, Nov 29 2014

%t nn = 14; Table[

%t Take[Drop[

%t CoefficientList[

%t Series[ Sum[

%t u z^k/(1 - u z^k) Product[1/(1 - z^i), {i, 1, k - 1}]^2, {k,

%t 1, nn}], {z, 0, nn}], {z, u}], 1], n, {2, n + 1}][[n]], {n,

%t 1, nn}] // Grid

%Y Columns k=1-10 give: A006330, A114921, A226541, A320315, A320316, A320317, A320318, A320319, A320320, A320321.

%Y Row sums give A001523.

%Y Main diagonal gives A000012.

%K nonn,tabl

%O 1,4

%A _Geoffrey Critzer_, Nov 29 2014