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A128249 T(n,k) is the number of unlabeled acyclic single-source automata with n transient states on a (k+1)-letter input alphabet.

%I A128249 #9 Apr 18 2024 11:22:40
%S A128249 1,3,1,16,7,1,127,139,15,1,1363,5711,1000,31,1,18628,408354,189035,
%T A128249 6631,63,1,311250,45605881,79278446,5470431,42196,127,1,6173791,
%U A128249 7390305396,63263422646,12703473581,147606627,262459,255,1,142190703,1647470410551
%N A128249 T(n,k) is the number of unlabeled acyclic single-source automata with n transient states on a (k+1)-letter input alphabet.
%C A128249 Table with rows n=1,2,... and columns k=1,2,3,... is read along antidiagonals.
%H A128249 David Callan, <a href="http://arXiv.org/abs/0704.0004">A determinant of Stirling Cycle Numbers Count Unlabeled Acyclic Single-Source Automata</a> math.CO/0704.0004.
%H A128249 Manosij Ghosh Dastidar and Michael Wallner, <a href="https://arxiv.org/abs/2404.08415">Asymptotics of relaxed k-ary trees</a>, arXiv:2404.08415 [math.CO], 2024. See p. 1.4.
%p A128249 T := proc(n,k) local kn,A,i,j ; kn := k*n ; A := matrix(kn,kn) ; for i from 1 to kn do for j from 1 to kn do A[i,j] := abs(combinat[stirling1](floor((i-1)/k)+2,floor((i-1)/k)+1+i-j)) ; od ; od ; linalg[det](A) ; end: for d from 1 to 9 do for n from d to 1 by -1 do k := d+1-n ; printf("%d, ",T(n,k)) ; od ; od;
%t A128249 t[n_, k_] := Module[{kn, a, i, j}, kn = k*n; For[i = 1, i <= kn, i++, For[j = 1, j <= kn, j++, a[i, j] = Abs[StirlingS1[Floor[(i-1)/k]+2, Max[0, Floor[(i-1)/k]+1+i-j]]]]]; Det[Array[a, {kn, kn}]]]; Table[t[n-k, k], {n, 1, 10}, {k, 1, n-1}] // Flatten (* _Jean-François Alcover_, Jan 10 2014, translated from Maple *)
%Y A128249 Cf. A082162, A082161.
%K A128249 nonn,tabl
%O A128249 1,2
%A A128249 _R. J. Mathar_, May 09 2007