A232539 Triangle read by rows: T(n,k) = number of partitions of n into at most four parts in which the largest part is equal to k, 0 <= k <= n.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 0, 2, 2, 1, 1, 0, 0, 2, 3, 2, 1, 1, 0, 0, 1, 3, 3, 2, 1, 1, 0, 0, 1, 3, 4, 3, 2, 1, 1, 0, 0, 0, 3, 4, 4, 3, 2, 1, 1, 0, 0, 0, 2, 5, 5, 4, 3, 2, 1, 1, 0, 0, 0, 1, 4, 6, 5, 4, 3, 2, 1, 1, 0, 0, 0, 1, 4, 6, 7, 5, 4, 3, 2, 1, 1

Offset: 0

Author

L. Edson Jeffery, Jan 02 2014

Keywords

• nonn
• tabl

Comments

Also number of partitions of n into k parts with parts in the range 1..4.

Examples

			Triangle T{n,k} begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1, 1;
  0, 1, 2, 1, 1;
  0, 0, 2, 2, 1, 1;
  0, 0, 2, 3, 2, 1, 1;
  0, 0, 1, 3, 3, 2, 1, 1;
  0, 0, 1, 3, 4, 3, 2, 1, 1;
  0, 0, 0, 3, 4, 4, 3, 2, 1, 1;
  ...
		

Links

• Louis Comtet, Advanced Combinatorics, Reidel (1974).

Crossrefs

Cf. A001400 (row sums), A219237, A233292 (row partial sums), A145362 (parts <=2), A339884 (parts <=3).

Programs

Maple

maxp := 4 :
gf := 1/mul(1-u*t^i,i=1..maxp) :
for n from 0 to 13 do
    for m from 0 to n do
        coeftayl(gf,t=0,n) ;
        coeftayl(%,u=0,m) ;
        printf("%d ",%);
    end do:
    printf("\n") ;
end do: # R. J. Mathar, May 27 2025

Formula

G.f.: 1/((1-u*t)*(1-u*t^2)*(1-u*t^3)*(1-u*t^4)). - [Comtet p. 97 [2c]]. - R. J. Mathar, May 27 2025