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A085771 Triangle read by rows. T(n, k) = A059438(n, k) for 1 <= k <= n, and T(n, 0) = n^0. 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

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

REFERENCES

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

LINKS

Table of n, a(n) for n=0..65.

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

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

EXAMPLE

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]

MAPLE

# Uses function PMatrix from A357368.

PMatrix(10, A003319); # Peter Luschny, Oct 09 2022

PROG

(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

CROSSREFS

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

Variants: A059439, A263484 (row reversed).

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

Sequence in context: A259790 A246654 A325111 * A253286 A344499 A284799

Adjacent sequences: A085768 A085769 A085770 * A085772 A085773 A085774

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Jul 22 2003

STATUS

approved

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Last modified February 3 01:16 EST 2023. Contains 360024 sequences. (Running on oeis4.)