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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.

%I A261719 #27 Dec 31 2020 19:42:25
%S A261719 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,
%T A261719 4232,4350,1602,0,15,654,6528,25100,44475,36744,11481,0,22,1577,23892,
%U A261719 136516,369675,512787,352793,95503,0,30,3560,80979,666800,2603670,5413842,6170486,3641992,871030
%N 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.
%C A261719 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.
%H A261719 Alois P. Heinz, <a href="/A261719/b261719.txt">Rows n = 0..140, flattened</a>
%F A261719 T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A261718(n,k-i).
%e A261719 A(3,2) = 12: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a.
%e A261719 Triangle T(n,k) begins:
%e A261719   1
%e A261719   0,  1;
%e A261719   0,  2,    3;
%e A261719   0,  3,   12,    10;
%e A261719   0,  5,   40,    81,     47;
%e A261719   0,  7,  104,   396,    544,    246;
%e A261719   0, 11,  279,  1751,   4232,   4350,   1602;
%e A261719   0, 15,  654,  6528,  25100,  44475,  36744,  11481;
%e A261719   0, 22, 1577, 23892, 136516, 369675, 512787, 352793, 95503;
%e A261719   ...
%p A261719 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A261719       b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))
%p A261719     end:
%p A261719 T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p A261719 seq(seq(T(n, k), k=0..n), n=0..10);
%t A261719 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 *)
%Y A261719 Columns k=0-10 give: A000007, A000041 (for n>0), A293366, A293367, A293368, A293369, A293370, A293371, A293372, A293373, A293374.
%Y A261719 Row sums give A035341.
%Y A261719 Main diagonal gives A005651.
%Y A261719 T(2n,n) gives A261732.
%Y A261719 Cf. A060642, A261718, A261781 (same for compositions).
%K A261719 nonn,tabl
%O A261719 0,5
%A A261719 _Alois P. Heinz_, Aug 29 2015