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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 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.)