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A143632 Table T(n,k), n>=0, k>=0, read by antidiagonals, where the e.g.f. for column k satisfies A_k(x) = exp(x*A_k(((x+1)^k-1)/k)) if k>0 and A_0(x) = exp(x*A_0(0)) = exp(x).

%I A143632 #16 Feb 24 2019 17:05:53
%S A143632 1,1,1,1,1,1,1,1,3,1,1,1,3,16,1,1,1,3,19,125,1,1,1,3,22,185,1296,1,1,
%T A143632 1,3,25,253,2541,16807,1,1,1,3,28,329,4256,45787,262144,1,1,1,3,31,
%U A143632 413,6471,96727,1037359,4782969,1,1,1,3,34,505,9216,175747,2828274,28649553,100000000,1
%N A143632 Table T(n,k), n>=0, k>=0, read by antidiagonals, where the e.g.f. for column k satisfies A_k(x) = exp(x*A_k(((x+1)^k-1)/k)) if k>0 and A_0(x) = exp(x*A_0(0)) = exp(x).
%H A143632 Alois P. Heinz, <a href="/A143632/b143632.txt">Antidiagonals n = 0..99, flattened</a>
%e A143632 Table begins:
%e A143632   1,    1,    1,    1,    1,    1, ...
%e A143632   1,    1,    1,    1,    1,    1, ...
%e A143632   1,    3,    3,    3,    3,    3, ...
%e A143632   1,   16,   19,   22,   25,   28, ...
%e A143632   1,  125,  185,  253,  329,  413, ...
%e A143632   1, 1296, 2541, 4256, 6471, 9216, ...
%p A143632 A:= proc(n,k) option remember; if n<=0 or k=0 then 1 else A(n-1,k)(((x+1)^k-1)/k) fi; unapply(convert(series(exp(x*%), x,n+1), polynom), x) end: T:= (n,k)-> coeff(A(n,k)(x), x,n)*n!: seq(seq(T(n,d-n), n=0..d), d=0..11);
%t A143632 a[n_, k_][x_] := Module[{f}, f = If[n <= 0 || k == 0, 1, a[n-1, k][((#+1)^k-1)/k]]&; Normal[Series[Exp[y*f[y]], {y, 0, n+1}]] /. y -> x]; t[n_, k_] := Coefficient[a[n, k][x], x, n]*n!; Table[t[n, d-n], {d, 0, 11}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 12 2014, translated from Maple *)
%Y A143632 Columns k=0-9 give: A000012, A000272, A143633, A143634, A143635, A143636, A143637, A143638, A143639, A143640.
%Y A143632 Main diagonal gives A306578.
%K A143632 nonn,tabl
%O A143632 0,9
%A A143632 _Alois P. Heinz_, Aug 27 2008