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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
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
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).
Sequence in context: A246654 A370506 A184182 * A253286 A344499 A284799
KEYWORD
easy,nonn,tabl
AUTHOR
Philippe Deléham, Jul 22 2003
STATUS
approved