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A144150
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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.
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24
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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, 35, 154, 358, 203, 1, 1, 1, 7, 51, 315, 1304, 2471, 877, 1, 1, 1, 8, 70, 561, 3455, 12915, 19302, 4140, 1, 1, 1, 9, 92, 910, 7556, 44590, 146115, 167894, 21147, 1, 1, 1, 10, 117
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OFFSET
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0,9
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COMMENTS
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A(n,k) is also the number of (k+1)-level labeled rooted trees with n leaves.
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
Equivalently, A(n,k) is the number of length k+1 multichains from bottom to top in the set partition lattice of an n-set. - Geoffrey Critzer, Dec 05 2020
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LINKS
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FORMULA
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E.g.f. of column k: 1 + g^(k+1)(x) with g = x-> exp(x)-1.
Column k+1 is Stirling transform of column k.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, ...
1, 5, 12, 22, 35, 51, ...
1, 15, 60, 154, 315, 561, ...
1, 52, 358, 1304, 3455, 7556, ...
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MAPLE
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g:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1
else (n-1)! *add(p(k)*b(n-k)/(k-1)!/(n-k)!, k=1..n) fi
end end:
A:= (n, k)-> (g@@k)(1)(n):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0 or k=0, 1,
add(binomial(n-1, j-1)*A(j, k-1)*A(n-j, k), j=1..n))
end:
# third Maple program:
b:= proc(n, t, m) option remember; `if`(t=0, 1, `if`(n=0,
b(m, t-1, 0), m*b(n-1, t, m)+b(n-1, t, m+1)))
end:
A:= (n, k)-> b(n, k, 0):
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MATHEMATICA
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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-François Alcover, Dec 06 2013 *)
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PROG
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(Python)
from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
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 range(1, n + 1)])
for n in range(51): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, Aug 07 2017
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CROSSREFS
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Columns k=0-10 give: A000012, A000110, A000258, A000307, A000357, A000405, A001669, A081624, A081629, A081697, A081740.
First lower diagonal gives A139383.
First upper diagonal gives A346802.
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KEYWORD
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AUTHOR
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STATUS
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approved
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