The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A153278 Array read by antidiagonals of higher order Fubini numbers. 2
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Mezo's abstract: The powers of matrices with Stirling number-coefficients are investigated. It is revealed that the elements of these matrices have a number of properties of the ordinary Stirling numbers. Moreover, "higher order" Bell, Fubini and Eulerian numbers can be defined. Hence we give a new interpretation for E. T. Bell's iterated exponential integers. In addition, it is worth to note that these numbers appear in combinatorial physics, in the problem of the normal ordering of quantum field theoretical operators. REFERENCES K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela, A. I. Solomon, Hierarchical Dobi'nski-type relations via substitution and the moment problem, J.Phys. A: Math.Gen. 37 3475-3487 (2004). LINKS Alois P. Heinz, Antidiagonals n = 1..101, flattened Istvan Mezo, On powers of Stirling matrices, Dec 21, 2008. 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.|.new p=5..|.1.|.7.|.71.|.949.|15775.|.313920.|.7279795.|.new =========================================================== MAPLE with(combinat): 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. 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 15 02:21 EDT 2021. Contains 343909 sequences. (Running on oeis4.)