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 A262071 Number T(n,k) of ordered partitions of an n-set with nondecreasing block sizes and maximal block size equal to k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 14
 1, 0, 1, 0, 2, 1, 0, 6, 3, 1, 0, 24, 18, 4, 1, 0, 120, 90, 30, 5, 1, 0, 720, 630, 200, 45, 6, 1, 0, 5040, 4410, 1610, 350, 63, 7, 1, 0, 40320, 37800, 13440, 3290, 560, 84, 8, 1, 0, 362880, 340200, 130200, 30870, 5922, 840, 108, 9, 1, 0, 3628800, 3515400, 1327200, 334950, 61992, 9870, 1200, 135, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA E.g.f. of column k: x^k * Product_{i=1..k} (i-1)!/(i!-x^i). EXAMPLE T(3,1) = 6: 1|2|3, 1|3|2, 2|1|3, 2|3|1, 3|1|2, 3|2|1. T(3,2) = 3: 1|23, 2|13, 3|12. T(3,3) = 1: 123. Triangle T(n,k) begins:   1;   0,     1;   0,     2,     1;   0,     6,     3,     1;   0,    24,    18,     4,    1;   0,   120,    90,    30,    5,   1;   0,   720,   630,   200,   45,   6,  1;   0,  5040,  4410,  1610,  350,  63,  7, 1;   0, 40320, 37800, 13440, 3290, 560, 84, 8, 1; MAPLE b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,        b(n, i-1)+`if`(i>n, 0, binomial(n, i)*b(n-i, i))))     end: T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)): seq(seq(T(n, k), k=0..n), n=0..12); MATHEMATICA b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, Binomial[n, i]*b[n - i, i]]]]; T[n_, k_] :=  b[n, k] - If[k == 0, 0, b[n, k - 1]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 12 2016, Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000007, A000142 (for n>0), A272492, A272493, A272494, A272495, A272496, A272497, A272498, A272499, A272500. Main diagonal gives A000012. Row sums give A005651. T(2n,n) gives A266518. Cf. A262072. Sequence in context: A066387 A180663 A301924 * A011312 A275328 A147720 Adjacent sequences:  A262068 A262069 A262070 * A262072 A262073 A262074 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 10 2015 STATUS approved

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Last modified January 22 09:44 EST 2019. Contains 319363 sequences. (Running on oeis4.)