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A261280 Number of ways to start with set {1,2,...,n} and then repeat n times: partition each set into subsets.

%I A261280 #31 Aug 04 2021 16:50:08
%S A261280 1,1,3,22,315,7556,274778,14140722,979687005,87998832685,
%T A261280 9951699489061,1384060090903535,232230523534594676,
%U A261280 46265730933522733556,10797461309089628151462,2918087323005280354349508,904185772556792011572372117,318432010852077710049833537040
%N A261280 Number of ways to start with set {1,2,...,n} and then repeat n times: partition each set into subsets.
%H A261280 Alois P. Heinz, <a href="/A261280/b261280.txt">Table of n, a(n) for n = 0..247</a>
%F A261280 a(n) = n! * [x^n] 1 + g^(k+1)(x), where g(x) = exp(x)-1.
%F A261280 From _Vaclav Kotesovec_, Aug 14 2015: (Start)
%F A261280 Conjecture: a(n) ~ c * n^(2*n-5/6) / (2^(n-1) * exp(n)), where c = 7.7889...
%F A261280 a(n) ~ exp(1) * A139383(n).
%F A261280 (End)
%e A261280 a(2) = 3: 12->12->12, 12->12->1|2, 12->1|2->1|2.
%e A261280 a(3) = 22: 123->123->123->123, 123->123->123->12|3, 123->123->123->1|23, 123->123->123->13|2, 123->123->123->1|2|3, 123->123->12|3->12|3, 123->123->12|3->1|2|3, 123->123->1|23->1|23, 123->123->1|23->1|2|3, 123->123->13|2->13|2, 123->123->13|2->1|2|3, 123->123->1|2|3->1|2|3, 123->12|3->12|3->12|3, 123->12|3->12|3->1|2|3, 123->12|3->1|2|3->1|2|3, 123->1|23->1|23->1|23, 123->1|23->1|23->1|2|3, 123->1|23->1|2|3->1|2|3, 123->13|2->13|2->13|2, 123->13|2->13|2->1|2|3, 123->13|2->1|2|3->1|2|3, 123->1|2|3->1|2|3->1|2|3.
%p A261280 g:= x-> exp(x)-1:
%p A261280 egf:= k-> 1+(g@@(k+1))(x):
%p A261280 a:= n-> n! * coeff(series(egf(n), x, n+1), x, n):
%p A261280 seq(a(n), n=0..20);
%p A261280 # second Maple program:
%p A261280 A:= proc(n, k) option remember; `if`(n=0 or k=0, 1,
%p A261280       add(binomial(n-1, j-1)*A(j, k-1)*A(n-j, k), j=1..n))
%p A261280     end:
%p A261280 a:= n-> A(n$2):
%p A261280 seq(a(n), n=0..20);
%p A261280 # third Maple program:
%p A261280 b:= proc(n, t, m) option remember; `if`(t=0, 1, `if`(n=0,
%p A261280       b(m, t-1, 0), m*b(n-1, t, m)+b(n-1, t, m+1)))
%p A261280     end:
%p A261280 a:= n-> b(n$2, 0):
%p A261280 seq(a(n), n=0..20);  # _Alois P. Heinz_, Aug 04 2021
%t A261280 Clear[t]; t[n_, k_]:=t[n, k] = If[n==0 || k==0, 1, Sum[Binomial[n-1, j-1]*t[j, k-1]*t[n-j, k], {j, 1, n}]]; Table[t[n, n], {n, 0, 20}] (* _Vaclav Kotesovec_, Aug 14 2015 after _Alois P. Heinz_ *)
%o A261280 (Python)
%o A261280 from sympy.core.cache import cacheit
%o A261280 from sympy import binomial
%o A261280 @cacheit
%o A261280 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))
%o A261280 def a(n): return A(n, n)
%o A261280 print([a(n) for n in range(21)]) # _Indranil Ghosh_, Aug 07 2017
%Y A261280 Main diagonal of A144150.
%Y A261280 Cf. A139383, A306187, A306188.
%K A261280 nonn
%O A261280 0,3
%A A261280 _Alois P. Heinz_, Aug 14 2015