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%I #29 Oct 09 2022 07:58:55
%S 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,
%T 3447,1142,327,92,25,6,1,0,29093,8411,2109,531,135,33,7,1,0,273343,
%U 69692,15366,3440,800,188,42,8,1,0,2829325,642581,125316,24892,5226,1146,252,52,9,1
%N Triangle read by rows. T(n, k) = A059438(n, k) for 1 <= k <= n, and T(n, 0) = n^0.
%C The convolution triangle of A003319, the number of irreducible permutations. - _Peter Luschny_, Oct 09 2022
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 262 (#14).
%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000056">The decomposition number of a permutation.</a>
%F 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.
%F 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.
%F 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
%e Triangle starts:
%e [0] [1]
%e [1] [0, 1]
%e [2] [0, 1, 1]
%e [3] [0, 3, 2, 1]
%e [4] [0, 13, 7, 3, 1]
%e [5] [0, 71, 32, 12, 4, 1]
%e [6] [0, 461, 177, 58, 18, 5, 1]
%e [7] [0, 3447, 1142, 327, 92, 25, 6, 1]
%e [8] [0, 29093, 8411, 2109, 531, 135, 33, 7, 1]
%e [9] [0, 273343, 69692, 15366, 3440, 800, 188, 42, 8, 1]
%p # Uses function PMatrix from A357368.
%p PMatrix(10, A003319); # _Peter Luschny_, Oct 09 2022
%o (SageMath) # Using function delehamdelta from A084938.
%o def A085771_triangle(n) :
%o a = [0, 1] + [(i + 3) // 2 for i in range(1, n-1)]
%o b = [0^i for i in range(n)]
%o return delehamdelta(a, b)
%o A085771_triangle(9) # _Peter Luschny_, Sep 10 2022
%Y T(2*n, n) = A308650(n).
%Y Variants: A059439, A263484 (row reversed).
%Y Cf. A003319, A059440, A055998, A059438, A000007, A084938.
%K easy,nonn,tabl
%O 0,8
%A _Philippe Deléham_, Jul 22 2003
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