A293366 -id:A293366 - cp's OEIS frontend

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.

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

Offset: 0

Alois P. Heinz, Aug 29 2015

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.

			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;
  ...

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

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).

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);

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-François Alcover, Jan 21 2017, translated from Maple *)

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

A256155 Decimal expansion of a constant related to A074141.

Original entry on oeis.org

1, 8, 5, 6, 3, 1, 4, 6, 5, 6, 3, 6, 1, 0, 1, 1, 4, 7, 2, 7, 4, 7, 5, 3, 5, 4, 2, 3, 2, 2, 6, 9, 2, 8, 4, 0, 4, 8, 4, 2, 5, 8, 8, 5, 9, 4, 7, 2, 2, 9, 0, 7, 0, 6, 8, 2, 0, 1, 0, 2, 5, 3, 9, 0, 7, 5, 4, 7, 6, 3, 7, 5, 6, 4, 7, 4, 6, 5, 1, 0, 0, 1, 9, 7, 1, 5, 7, 0, 7, 2, 9, 6, 0, 4, 2, 1, 5, 1, 9, 3, 0, 3, 9, 4, 5, 1, 9

Offset: 2

Vaclav Kotesovec, Mar 16 2015

			18.5631465636101147274753542322692840484258859472290706820102539...

Vaclav Kotesovec, Table of n, a(n) for n = 2..285

Cf. A074141, A293366.

Equals limit n->infinity A074141(n) / 2^n.

Equals Product_{k>=2} 1/(1-(k+1)/2^k). - Vaclav Kotesovec, May 10 2021

cp's OEIS Frontend

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.

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