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A143565 Triangle T(n,k), n>=1, 1<=k<=n, where the e.g.f. for column k satisfies: A_k(x) = exp(x*A_k(x^k/k!)).

%I A143565 #24 Nov 29 2023 10:31:22
%S A143565 1,3,1,16,4,1,125,13,5,1,1296,46,21,6,1,16807,241,61,31,7,1,262144,
%T A143565 1471,211,106,43,8,1,4782969,9409,1401,281,169,57,9,1,100000000,67348,
%U A143565 8065,946,505,253,73,10,1,2357947691,564841,37241,7561,1261,841,361,91,11,1
%N A143565 Triangle T(n,k), n>=1, 1<=k<=n, where the e.g.f. for column k satisfies: A_k(x) = exp(x*A_k(x^k/k!)).
%H A143565 Alois P. Heinz, <a href="/A143565/b143565.txt">Rows n = 1..100, flattened</a>
%F A143565 E.g.f. for column k satisfies: A_k(x) = exp(x*A_k(x^k/k!)).
%F A143565 T(0,k) = 1; T(n,k) = (n-1)! * Sum_{j=0..floor((n-1)/k)} (k*j+1) * T(j,k) * T(n-1-k*j,k) / (k!^j * j! * (n-1-k*j)!). - _Seiichi Manyama_, Nov 28 2023
%e A143565 Triangle begins:
%e A143565        1;
%e A143565        3,    1;
%e A143565       16,    4,    1;
%e A143565      125,   13,    5,    1;
%e A143565     1296,   46,   21,    6,    1;
%e A143565    16807,  241,   61,   31,    7,    1;
%e A143565   262144, 1471,  211,  106,   43,    8,    1;
%e A143565   ...
%p A143565 A:= proc(n,k::posint) option remember; if n<=0 then 1 else unapply(
%p A143565       convert(series(exp(x*A(n-k,k)(x^k/k!)), x,n+1), polynom),x) fi
%p A143565     end:
%p A143565 T:= (n,k)-> coeff(A(n,k)(x), x,n)*n!:
%p A143565 seq(seq(T(n,k), k=1..n), n=1..12);
%t A143565 a[n_, k_] := a[n, k] = If[n <= 0, 1&, Function[x, Series[E^(x*a[n - k, k][x^k/k!]), {x, 0, n+1}] // Normal // Evaluate]]; t[n_, k_] := Coefficient[a[n, k][x], x, n]*n!; Flatten[Table[Table[t[n, k], {k, 1, n}], {n, 1, 12}]] (* _Jean-François Alcover_, Dec 16 2013, translated from Maple *)
%Y A143565 Columns 1-9: A000272, A143566, A143567, A143568, A143569, A143570, A143571, A143572, A143573.
%Y A143565 T(2n,n) gives A319924.
%K A143565 nonn,tabl
%O A143565 1,2
%A A143565 _Alois P. Heinz_, Aug 24 2008