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A216350 Triangle T(n,k) in which n-th row lists in increasing order the values of the n-th derivative at x=1 of all functions that are representable as x^x^...^x with n x's and parentheses inserted in all possible ways; n>=1, 1<=k<=A000081(n).

%I A216350 #16 Mar 30 2016 15:14:08
%S A216350 1,2,9,12,56,80,100,156,480,660,890,950,1180,1360,1420,1880,3160,5094,
%T A216350 6534,8874,10848,10974,13014,13314,14928,14988,15114,20268,21474,
%U A216350 22008,24042,29682,31968,34974,35382,50496,87990,65534,78134,102494,131684,141974
%N A216350 Triangle T(n,k) in which n-th row lists in increasing order the values of the n-th derivative at x=1 of all functions that are representable as x^x^...^x with n x's and parentheses inserted in all possible ways; n>=1, 1<=k<=A000081(n).
%H A216350 Alois P. Heinz, <a href="/A216350/b216350.txt">Rows n = 1..12, flattened</a>
%e A216350 For n=4 the A000081(4) = 4 functions and their 4th derivatives at x=1 are x^(x^3)->156, x^(x^x*x)->100, x^(x^(x^2))->80, x^(x^(x^x))->56 => 4th row = [56, 80, 100, 156].
%e A216350 Triangle T(n,k) begins:
%e A216350 :    1;
%e A216350 :    2;
%e A216350 :    9,   12;
%e A216350 :   56,   80,  100,   156;
%e A216350 :  480,  660,  890,   950,  1180,  1360,  1420,  1880,  3160;
%e A216350 : 5094, 6534, 8874, 10848, 10974, 13014, 13314, 14928, 14988, 15114, ...
%p A216350 with(combinat):
%p A216350 F:= proc(n) F(n):= `if`(n<2, [x$n], map(h->x^h, g(n-1, n-1))) end:
%p A216350 g:= proc(n, i) option remember; `if`(n=0 or i=1, [x^n],
%p A216350      `if`(i<1, [], [seq(seq(seq(mul(F(i)[w[t]-t+1], t=1..j)*v,
%p A216350       w=choose([$1..nops(F(i))+j-1], j)), v=g(n-i*j, i-1)), j=0..n/i)]))
%p A216350     end:
%p A216350 T:= n-> sort(map(f-> n!*coeff(series(subs(x=x+1, f)
%p A216350                  , x, n+1), x, n), F(n)))[]:
%p A216350 seq(T(n), n=1..7);
%Y A216350 First column gives: A033917.
%Y A216350 Last elements of rows give: A216351.
%Y A216350 A version with different ordering of row elements is: A216349.
%Y A216350 Rows sums give: A216281.
%Y A216350 Cf. A000081, A215703.
%K A216350 nonn,tabf
%O A216350 1,2
%A A216350 _Alois P. Heinz_, Sep 04 2012