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A261718 Number A(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; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A261718 #19 Sep 03 2018 19:51:11
%S A261718 1,1,0,1,1,0,1,2,2,0,1,3,7,3,0,1,4,15,18,5,0,1,5,26,55,50,7,0,1,6,40,
%T A261718 124,216,118,11,0,1,7,57,235,631,729,301,15,0,1,8,77,398,1470,2780,
%U A261718 2621,684,22,0,1,9,100,623,2955,8001,12954,8535,1621,30,0
%N A261718 Number A(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; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A261718 Alois P. Heinz, <a href="/A261718/b261718.txt">Antidiagonals n = 0..140, flattened</a>
%F A261718 A(n,k) = Sum_{i=0..k} C(k,i) * A261719(n,k-i).
%e A261718 A(3,2) = 18: 3aaa, 3aab, 3abb, 3bbb, 2aa1a, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 2bb1b, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a, 1b1b1b.
%e A261718 Square array A(n,k) begins:
%e A261718   1,  1,   1,    1,     1,      1,      1,       1, ...
%e A261718   0,  1,   2,    3,     4,      5,      6,       7, ...
%e A261718   0,  2,   7,   15,    26,     40,     57,      77, ...
%e A261718   0,  3,  18,   55,   124,    235,    398,     623, ...
%e A261718   0,  5,  50,  216,   631,   1470,   2955,    5355, ...
%e A261718   0,  7, 118,  729,  2780,   8001,  19158,   40299, ...
%e A261718   0, 11, 301, 2621, 12954,  45865, 130453,  317905, ...
%e A261718   0, 15, 684, 8535, 55196, 241870, 820554, 2323483, ...
%p A261718 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A261718       b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))
%p A261718     end:
%p A261718 A:= (n, k)-> b(n, n, k):
%p A261718 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A261718 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]]]]; A[n_, k_] := b[n, n, k]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Feb 22 2016, after _Alois P. Heinz_ *)
%Y A261718 Columns k=0-10 give: A000007, A000041, A074141, A261737, A261738, A261739, A261740, A261741, A261742, A261743, A261744.
%Y A261718 Rows n=0-2 give: A000012, A001477, A005449.
%Y A261718 Main diagonal gives A209668.
%Y A261718 Cf. A144064, A261719, A261780.
%K A261718 nonn,tabl
%O A261718 0,8
%A A261718 _Alois P. Heinz_, Aug 29 2015