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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.

%I A209631 #16 Feb 27 2014 03:43:40
%S A209631 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,
%T A209631 7,49,280,967,1057,7,1,1,8,68,518,2883,8549,6322,8,1,1,9,90,859,6689,
%U A209631 34817,85829,41393,9,1,1,10,115,1321,13310,101841,481477
%N 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.
%C A209631 Motivation: The exponential transform applied n times to the constant function 1 evaluated at k was studied by E. T. Bell (see A144150).
%H A209631 Discussion on seqcomp: <a href="http://groups.google.com/group/seqcomp/browse_thread/thread/c8c8725327093f95">A little challenge</a>
%e A209631 n\k [0][1][2] [3]   [4]    [5]     [6]
%e A209631 [0]  0, 1, 2,  3,    4,     5,      6
%e A209631 [1]  1, 1, 3, 10,   41,   196,   1057   [A000248]
%e A209631 [2]  1, 1, 4, 20,  127,   967,   8549   [A007550]
%e A209631 [3]  1, 1, 5, 33,  280,  2883,  34817
%e A209631 [4]  1, 1, 6, 49,  518,  6689, 101841
%e A209631 [5]  1, 1, 7, 68,  859, 13310, 243946
%e A209631 [6]  1, 1, 8, 90, 1321, 23851, 510502
%e A209631 column3(n) = (3*n^2 + 11*n + 6)/2!
%e A209631 column4(n) = (18*n^3 + 93*n^2 + 111*n + 24)/3!
%e A209631 column5(n) = (180*n^4 + 1180*n^3 + 2160*n^2 + 1064*n + 120)/4!
%e A209631 column6(n) = (2700*n^5+21225*n^4+51850*n^3+41835*n^2+8510*n+720)/5!
%p A209631 # Implementation after Alois P. Heinz.
%p A209631 exptr := proc(p) local g; g := proc(n) option remember; local k;
%p A209631 `if`(n=0, 1, add(binomial(n-1, k-1)*p(k)*g(n-k), k=1..n)) end end:
%p A209631 A209631 := (n,k) -> (exptr@@n)(m->m)(k):
%p A209631 seq(lprint(seq(A209631(n,k), k=0..6)), n=0..6);
%t A209631 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_ *)
%Y A209631 Cf. A000248, A007550, A111672, A144150.
%K A209631 nonn,tabl
%O A209631 0,6
%A A209631 _Peter Luschny_, Mar 11 2012