A363009 -id:A363009 - cp's OEIS frontend

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

Jonathan Vos Post, Dec 22 2008

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

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).

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

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

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 *)

More terms from Alois P. Heinz, Feb 02 2009

A363007 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - f^k(x)), where f(x) = exp(x) - 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 13, 24, 1, 1, 5, 23, 75, 120, 1, 1, 6, 36, 175, 541, 720, 1, 1, 7, 52, 342, 1662, 4683, 5040, 1, 1, 8, 71, 594, 4048, 18937, 47293, 40320, 1, 1, 9, 93, 949, 8444, 57437, 251729, 545835, 362880, 1, 1, 10, 118, 1425, 15775, 143783, 950512, 3824282, 7087261, 3628800

Offset: 0

Seiichi Manyama, May 12 2023

			Square array begins:
    1,   1,    1,    1,    1,     1, ...
    1,   1,    1,    1,    1,     1, ...
    2,   3,    4,    5,    6,     7, ...
    6,  13,   23,   36,   52,    71, ...
   24,  75,  175,  342,  594,   949, ...
  120, 541, 1662, 4048, 8444, 15775, ...

Columns k=0..5 give A000142, A000670, A083355, A099391, A363008, A363009.
Main diagonal gives A363010.
Cf. A153278, A351420, A351429.

T(n, k) = if(k==0, n!, sum(j=0, n, stirling(n, j, 2)*T(j, k-1)));

T(n,k) = Sum_{j=0..n} Stirling2(n,j) * T(j,k-1), k>1, T(n,0) = n!.

T(n,k) = A153278(k,n) for n >= 1 and k >= 1.

cp's OEIS Frontend

A153278 Array read by antidiagonals of higher order Fubini numbers.

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