A384993 Triangle read by rows: T(n,k) is the number of compositions (p_0,p_1,...,p_m) of n such that max(i + p_i) = k for 0 <= i <= m.

1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 4, 4, 0, 0, 0, 3, 8, 5, 0, 0, 0, 1, 12, 13, 6, 0, 0, 0, 0, 12, 26, 19, 7, 0, 0, 0, 0, 8, 40, 46, 26, 8, 0, 0, 0, 0, 4, 48, 88, 73, 34, 9, 0, 0, 0, 0, 1, 47, 140, 163, 108, 43, 10, 0, 0, 0, 0, 0, 38, 190, 307, 273, 152, 53, 11

Offset: 0

Author

John Tyler Rascoe, Sep 01 2025

Keywords

• nonn
• easy
• tabl
• new

Examples

			Triangle begins:
    k=0  1  2  3   4   5   6   7  8
 n=0 [1]
 n=1 [0, 1]
 n=2 [0, 0, 2]
 n=3 [0, 0, 1, 3]
 n=4 [0, 0, 0, 4,  4]
 n=5 [0, 0, 0, 3,  8,  5]
 n=6 [0, 0, 0, 1, 12, 13,  6]
 n=7 [0, 0, 0, 0, 12, 26, 19,  7]
 n=8 [0, 0, 0, 0,  8, 40, 46, 26, 8]
...
The composition of n = 8, (2,1,3,1,1) has values of i + p_i : 2,2,5,4,5 that have a maximum value of 5 so this composition is counted under T(8,5) = 40.
T(4,3) = 4 counts: (1,2,1), (2,1,1), (2,2), (3,1).
T(4,4) = 4 counts: (1,1,1,1), (1,1,2), (1,3), (4).
		

Crossrefs

Cf. A008930, A238859.

Row sums A011782, empirical column sums A001339, empirical second diagonal A034856.

Programs

PARI

r(i) = {sum(k=1,i, x^k)}
P(n) = {1 + sum(i=1,n, prod(j=0,i-1, r(n-j)))}
C(k) = {sum(i=1,k, x^k * P(k-i) * prod(j=0,i-2, 1 + r(k-j-2)))}
T_rowlist(max_row) = {my(N = max_row+1, h = 1 + sum(k=1,N, y^k * C(k))); vector(N, n, Vecrev(polcoeff(h, n-1)))}