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A209631 Square array A(n,k), n>=0, k>=0, read by antidiagonals, A(n,k) = exponential transform applied n times to identity function, evaluated at k. 2
0, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 10, 4, 1, 1, 5, 20, 41, 5, 1, 1, 6, 33, 127, 196, 6, 1, 1, 7, 49, 280, 967, 1057, 7, 1, 1, 8, 68, 518, 2883, 8549, 6322, 8, 1, 1, 9, 90, 859, 6689, 34817, 85829, 41393, 9, 1, 1, 10, 115, 1321, 13310, 101841, 481477 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Motivation: The exponential transform applied n times to the constant function 1 evaluated at k was studied by E. T. Bell (see A144150).

LINKS

Table of n, a(n) for n=0..62.

Discussion on seqcomp: A little challenge

EXAMPLE

n\k [0][1][2] [3]   [4]    [5]     [6]

[0]  0, 1, 2,  3,    4,     5,      6

[1]  1, 1, 3, 10,   41,   196,   1057   [A000248]

[2]  1, 1, 4, 20,  127,   967,   8549   [A007550]

[3]  1, 1, 5, 33,  280,  2883,  34817

[4]  1, 1, 6, 49,  518,  6689, 101841

[5]  1, 1, 7, 68,  859, 13310, 243946

[6]  1, 1, 8, 90, 1321, 23851, 510502

column3(n) = (3*n^2 + 11*n + 6)/2!

column4(n) = (18*n^3 + 93*n^2 + 111*n + 24)/3!

column5(n) = (180*n^4 + 1180*n^3 + 2160*n^2 + 1064*n + 120)/4!

column6(n) = (2700*n^5+21225*n^4+51850*n^3+41835*n^2+8510*n+720)/5!

MAPLE

# Implementation after Alois P. Heinz.

exptr := proc(p) local g; g := proc(n) option remember; local k;

`if`(n=0, 1, add(binomial(n-1, k-1)*p(k)*g(n-k), k=1..n)) end end:

A209631 := (n, k) -> (exptr@@n)(m->m)(k):

seq(lprint(seq(A209631(n, k), k=0..6)), n=0..6);

MATHEMATICA

exptr[p_] := Module[{g}, g[n_] := g[n] = Module[{k}, If[n == 0, 1, Sum[Binomial[n-1, k-1]*p[k]*g[n-k], {k, 1, n}]]]; g]; A209631[n_, k_] := Nest[exptr, Identity, n][k]; Table[A209631[n-k , k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Feb 27 2014, after Alois P. Heinz *)

CROSSREFS

Cf. A000248, A007550, A111672, A144150.

Sequence in context: A099597 A283113 A123610 * A309876 A059922 A229556

Adjacent sequences:  A209628 A209629 A209630 * A209632 A209633 A209634

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Mar 11 2012

STATUS

approved

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Last modified February 25 08:33 EST 2021. Contains 341606 sequences. (Running on oeis4.)