A247255 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.

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, 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, 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

Offset: 1

Author

Geoffrey Critzer, Nov 29 2014

Keywords

• nonn
• tabl

Comments

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

Examples

			    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, 0, 0, 0, 1;
  106, 16, 5, 2, 0, 0, 0, 0, 1;
  170, 27, 7, 3, 1, 0, 0, 0, 0, 1;
		

References

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

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

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

Row sums give A001523.

Main diagonal gives A000012.

Programs

Maple

b:= proc(n, i) option remember; local r; expand(
      `if`(i>n, 0, `if`(irem(n, i, 'r')=0, x^r, 0)+
      add(b(n-i*j, i+1)*(j+1), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 1)):
seq(T(n), n=1..14);  # Alois P. Heinz, Nov 29 2014

Mathematica

nn = 14; Table[
  Take[Drop[
     CoefficientList[
      Series[ Sum[
        u z^k/(1 - u z^k) Product[1/(1 - z^i), {i, 1, k - 1}]^2, {k,
         1, nn}], {z, 0, nn}], {z, u}], 1], n, {2, n + 1}][[n]], {n,
   1, nn}] // Grid

Formula

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

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