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A261719 Number T(n,k) of partitions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order and all k letters occur at least once in the partition; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 16
1, 0, 1, 0, 2, 3, 0, 3, 12, 10, 0, 5, 40, 81, 47, 0, 7, 104, 396, 544, 246, 0, 11, 279, 1751, 4232, 4350, 1602, 0, 15, 654, 6528, 25100, 44475, 36744, 11481, 0, 22, 1577, 23892, 136516, 369675, 512787, 352793, 95503, 0, 30, 3560, 80979, 666800, 2603670, 5413842, 6170486, 3641992, 871030 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

T(n,k) is defined for n,k >= 0.  The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

FORMULA

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A261718(n,k-i).

EXAMPLE

A(3,2) = 12: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a.

Triangle T(n,k) begins:

  1

  0,  1;

  0,  2,    3;

  0,  3,   12,    10;

  0,  5,   40,    81,     47;

  0,  7,  104,   396,    544,    246;

  0, 11,  279,  1751,   4232,   4350,   1602;

  0, 15,  654,  6528,  25100,  44475,  36744,  11481;

  0, 22, 1577, 23892, 136516, 369675, 512787, 352793, 95503;

MAPLE

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

      b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))

    end:

T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):

seq(seq(T(n, k), k=0..n), n=0..10);

MATHEMATICA

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k]*Binomial[i + k - 1, k - 1]]]]; T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jan 21 2017, translated from Maple *)

CROSSREFS

Columns k=0-10 give: A000007, A000041 (for n>0), A293366, A293367, A293368, A293369, A293370, A293371, A293372, A293373, A293374.

Row sums give A035341.

Main diagonal gives A005651.

T(2n,n) gives A261732.

Cf. A060642, A261718, A261781 (same for compositions).

Sequence in context: A194365 A216217 A253283 * A137663 A257740 A161628

Adjacent sequences:  A261716 A261717 A261718 * A261720 A261721 A261722

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 29 2015

STATUS

approved

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Last modified October 14 02:29 EDT 2019. Contains 327995 sequences. (Running on oeis4.)