A085771 Triangle read by rows. T(n, k) = A059438(n, k) for 1 <= k <= n, and T(n, 0) = n^0.

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 13, 7, 3, 1, 0, 71, 32, 12, 4, 1, 0, 461, 177, 58, 18, 5, 1, 0, 3447, 1142, 327, 92, 25, 6, 1, 0, 29093, 8411, 2109, 531, 135, 33, 7, 1, 0, 273343, 69692, 15366, 3440, 800, 188, 42, 8, 1, 0, 2829325, 642581, 125316, 24892, 5226, 1146, 252, 52, 9, 1

Offset: 0

Author

Philippe Deléham, Jul 22 2003

Keywords

• easy
• nonn
• tabl

Comments

The convolution triangle of A003319, the number of irreducible permutations. - Peter Luschny, Oct 09 2022

Examples

			Triangle starts:
[0] [1]
[1] [0,      1]
[2] [0,      1,     1]
[3] [0,      3,     2,     1]
[4] [0,     13,     7,     3,    1]
[5] [0,     71,    32,    12,    4,   1]
[6] [0,    461,   177,    58,   18,   5,   1]
[7] [0,   3447,  1142,   327,   92,  25,   6,  1]
[8] [0,  29093,  8411,  2109,  531, 135,  33,  7, 1]
[9] [0, 273343, 69692, 15366, 3440, 800, 188, 42, 8, 1]
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 262 (#14).

Links

• FindStat - Combinatorial Statistic Finder, The decomposition number of a permutation.

Crossrefs

T(2*n, n) = A308650(n).

Variants: A059439, A263484 (row reversed).

Cf. A003319, A059440, A055998, A059438, A000007, A084938.

Programs

Maple

# Uses function PMatrix from A357368.
PMatrix(10, A003319); # Peter Luschny, Oct 09 2022

SageMath

# Using function delehamdelta from A084938.
def A085771_triangle(n) :
    a = [0, 1] + [(i + 3) // 2 for i in range(1, n-1)]
    b = [0^i for i in range(n)]
    return delehamdelta(a, b)
A085771_triangle(9) # Peter Luschny, Sep 10 2022

Formula

Let f(x) = Sum_{n>=0} n!*x^n, g(x) = 1 - 1/f(x). Then g(x) is the g.f. of the second column, A003319.

Triangle T(n, k) read by rows, given by [0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA A000007, where DELTA is Deléham's operator defined in A084938.

G.f.: 1/(1 - xy/(1 - x/(1 - 2x/(1 - 2x/(1 - 3x/(1 - 3x/(1 - 4x/(1-.... (continued fraction). - Paul Barry, Jan 29 2009