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

1,3

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).

Example

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
===========================================================

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 *)

Crossrefs

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

Sequence in context: A010756 A191857 A248044 * A010284 A095328 A066712

Adjacent sequences: A153275 A153276 A153277 * A153279 A153280 A153281

Keyword

easy,nonn,tabl

Author

Jonathan Vos Post, Dec 22 2008

Extensions

More terms from Alois P. Heinz, Feb 02 2009

Status

approved