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.

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

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-François Alcover, Feb 27 2014, after Alois P. Heinz *)

Crossrefs

Cf. A000248, A007550, A111672, A144150.

Sequence in context: A358146 A283113 A123610 * A309876 A059922 A229556

Adjacent sequences: A209628 A209629 A209630 * A209632 A209633 A209634

Keyword

nonn,tabl

Author

Peter Luschny, Mar 11 2012

Status

approved