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A144150 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where the e.g.f. of column k is 1+g^(k+1)(x) with g = x-> exp(x)-1. 22

%I

%S 1,1,1,1,1,1,1,1,2,1,1,1,3,5,1,1,1,4,12,15,1,1,1,5,22,60,52,1,1,1,6,

%T 35,154,358,203,1,1,1,7,51,315,1304,2471,877,1,1,1,8,70,561,3455,

%U 12915,19302,4140,1,1,1,9,92,910,7556,44590,146115,167894,21147,1,1,1,10,117

%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where the e.g.f. of column k is 1+g^(k+1)(x) with g = x-> exp(x)-1.

%C A(n,k) is also the number of (k+1)-level labeled rooted trees with n leaves.

%C Number of ways to start with set {1,2,...,n} and then repeat k times: partition each set into subsets. - _Alois P. Heinz_, Aug 14 2015

%H Alois P. Heinz, <a href="/A144150/b144150.txt">Antidiagonals n = 0..140, flattened</a>

%H E. T. Bell, <a href="http://doi.org/10.2307/1968633">The Iterated Exponential Integers</a>, Annals of Mathematics, 39(3) (1938), 539-557.

%H Pierpaolo Natalini, Paolo Emilio Ricci, <a href="https://doi.org/10.2298/AADM1702327N">Higher order Bell polynomials and the relevant integer sequences</a>, in Appl. Anal. Discrete Math. 11 (2017), 327-339.

%H Pierpaolo Natalini, Paolo E. Ricci, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Ricci/ricci3.html">Integer Sequences Connected with Extensions of the Bell Polynomials</a>, Journal of Integer Sequences, 2017, Vol. 20, #17.10.2.

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F E.g.f. of column k: 1 + g^(k+1)(x) with g = x-> exp(x)-1.

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 3, 4, 5, 6, ...

%e 1, 5, 12, 22, 35, 51, ...

%e 1, 15, 60, 154, 315, 561, ...

%e 1, 52, 358, 1304, 3455, 7556, ...

%p g:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1

%p else (n-1)! *add(p(k)*b(n-k)/(k-1)!/(n-k)!, k=1..n) fi

%p end end:

%p A:= (n,k)-> (g@@k)(1)(n):

%p seq(seq(A(n, d-n), n=0..d), d=0..12);

%p # second Maple program:

%p A:= proc(n, k) option remember; `if`(n=0 or k=0, 1,

%p add(binomial(n-1, j-1)*A(j, k-1)*A(n-j, k), j=1..n))

%p end:

%p seq(seq(A(n, d-n), n=0..d), d=0..12); # _Alois P. Heinz_, Aug 14 2015

%t g[k_] := g[k] = Nest[Function[x, E^x - 1], x, k]; a[n_, k_] := SeriesCoefficient[1 + g[k + 1], {x, 0, n}]*n!; Table[a[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* _Jean-Fran├žois Alcover_, Dec 06 2013 *)

%o (Python)

%o from sympy.core.cache import cacheit

%o from sympy import binomial

%o @cacheit

%o def A(n, k): return 1 if n==0 or k==0 else sum([binomial(n - 1, j - 1)*A(j, k - 1)*A(n - j, k) for j in xrange(1, n + 1)])

%o for n in xrange(51): print[A(k, n - k) for k in xrange(n + 1)] # _Indranil Ghosh_, Aug 07 2017

%Y Columns k=0-10 give: A000012, A000110, A000258, A000307, A000357, A000405, A001669, A081624, A081629, A081697, A081740.

%Y Rows n=0+1, 2-4 give: A000012, A000027, A000326, A005945.

%Y First lower diagonal gives A139383.

%Y Main diagonal gives A261280.

%Y Cf. A000142, A111672, A290353.

%K nonn,tabl

%O 0,9

%A _Alois P. Heinz_, Sep 11 2008

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Last modified March 18 13:47 EDT 2019. Contains 321289 sequences. (Running on oeis4.)