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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
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
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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Mar 11 2012
STATUS
approved