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

Author

Peter Luschny, Mar 11 2012

Keywords

• nonn
• tabl

Comments

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

Examples

			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!
		

Links

• Discussion on seqcomp: A little challenge

Crossrefs

Cf. A000248, A007550, A111672, A144150.

Programs

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 *)