A153278 Array read by antidiagonals of higher order Fubini numbers.

1, 1, 3, 1, 4, 13, 1, 5, 23, 75, 1, 6, 36, 175, 541, 1, 7, 52, 342, 1662, 4683, 1, 8, 71, 594, 4048, 18937, 47293, 1, 9, 93, 949, 8444, 57437, 251729, 545835, 1, 10, 118, 1425, 15775, 143783, 950512, 3824282, 7087261, 1, 11, 146, 2040, 27146, 313920, 2854261, 17975438, 65361237, 102247563

Offset: 1

Author

Jonathan Vos Post, Dec 22 2008

Keywords

• easy
• nonn
• tabl

Examples

			The table on p.6 of Mezo begins:
===========================================================
F_p,n|n=1|n=2|n=3.|.n=4.|..n=5.|....n=6.|.....n=7.|comment
===========================================================
p=1..|.1.|.3.|.13.|..75.|..541.|...4683.|...47293.|.A000670
p=2..|.1.|.4.|.23.|.175.|.1662.|..18937.|..251729.|.A083355
p=3..|.1.|.5.|.36.|.342.|.4048.|..57437.|..950512.|.A099391
p=4..|.1.|.6.|.52.|.594.|.8444.|.143783.|.2854261.|.A363008
p=5..|.1.|.7.|.71.|.949.|15775.|.313920.|.7279795.|.A363009
===========================================================
		

Links

• Alois P. Heinz, Antidiagonals n = 1..150, flattened
• Istvan Mezo, On powers of Stirling matrices, arXiv:0812.4047 [math.CO], 2008.
• K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela, and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem, arXiv:quant-ph/0312202, 2003; J.Phys. A: Math.Gen. 37 3475-3487 (2004).

Crossrefs

Cf. A000670, A083355, A099391, A153277, A363008, A363009.

Programs

Maple

f:= proc(n) option remember; local k; if n<=1 then 1 else
       add(binomial(n, k) *f(n-k), k=1..n) fi
    end:
stirtr:= proc(a) proc(n) option remember;
           add( a(k) *Stirling2(n,k), k=0..n)
         end end:
F:= (p,n)-> (stirtr@@(p-1))(f)(n):
seq(seq(F(d-n, n), n=1..d-1), d=1..13); # Alois P. Heinz, Feb 02 2009

Mathematica

f[n_] := f[n] = If[n <= 1, 1, Sum[Binomial[n, k]*f[n-k], {k, 1, n}]];
stirtr[a_] := Module[{g}, g[n_] := g[n] = Sum[a[k]*StirlingS2[n, k], {k, 0, n}]; g];
F[p_, n_] := (Composition @@ Table[stirtr, {p-1}])[f][n];
Table[Table[F[d-n, n], {n, 1, d-1}], {d, 1, 13}] // Flatten (* Jean-François Alcover, Mar 30 2016, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Feb 02 2009