A293367 -id:A293367 - 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).

A261737 Number of partitions of n where each part i is marked with a word of length i over a ternary alphabet whose letters appear in alphabetical order.

Original entry on oeis.org

1, 3, 15, 55, 216, 729, 2621, 8535, 28689, 91749, 296538, 929712, 2939063, 9093255, 28257123, 86681608, 266368959, 811501848, 2475331535, 7505567037, 22772955015, 68828023329, 208079886258, 627418618533, 1892181244828, 5696253823476, 17149663331259

Offset: 0

Alois P. Heinz, Aug 30 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=3 of A261718.

Cf. A293367.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i>n, 0, b(n-i, i)*binomial(i+2, 2))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]*Binomial[i + 2, 2]]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 24 2018, translated from Maple *)

a(n) ~ c * 3^n, where c = Product_{k>=2} 1/(1 - (k+1)*(k+2)/(2*3^k)) = 6.84620607349852135789816336867607014231681538613599316638081993041973716978... . - Vaclav Kotesovec, Nov 15 2016, updated May 10 2021

G.f.: Product_{k>=1} 1 / (1 - binomial(k+2,2)*x^k). - Ilya Gutkovskiy, May 09 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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