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A216349 Triangle T(n,k) in which n-th row lists 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 A216349 #19 Mar 30 2016 15:13:05
%S A216349 1,2,12,9,156,100,80,56,3160,1880,1180,1420,950,1360,890,660,480,
%T A216349 87990,50496,29682,35382,24042,22008,14928,31968,20268,14988,10848,
%U A216349 34974,21474,13314,15114,10974,13014,8874,6534,5094,3218628,1806476,1021552,588756,1189132
%N A216349 Triangle T(n,k) in which n-th row lists 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).
%C A216349 The ordering of the functions is the same as in A215703 and is defined by the algorithm below.
%H A216349 Alois P. Heinz, <a href="/A216349/b216349.txt">Rows n = 1..12, flattened</a>
%e A216349 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.
%e A216349 Triangle T(n,k) begins:
%e A216349 :     1;
%e A216349 :     2;
%e A216349 :    12,     9;
%e A216349 :   156,   100,    80,    56;
%e A216349 :  3160,  1880,  1180,  1420,   950,  1360,   890,   660,   480;
%e A216349 : 87990, 50496, 29682, 35382, 24042, 22008, 14928, 31968, 20268, ...
%p A216349 with(combinat):
%p A216349 F:= proc(n) F(n):= `if`(n<2, [x$n], map(h->x^h, g(n-1, n-1))) end:
%p A216349 g:= proc(n, i) option remember; `if`(n=0 or i=1, [x^n],
%p A216349      `if`(i<1, [], [seq(seq(seq(mul(F(i)[w[t]-t+1], t=1..j)*v,
%p A216349       w=choose([$1..nops(F(i))+j-1], j)), v=g(n-i*j, i-1)), j=0..n/i)]))
%p A216349     end:
%p A216349 T:= n-> map(f-> n!*coeff(series(subs(x=x+1, f), x, n+1), x, n), F(n))[]:
%p A216349 seq(T(n), n=1..7);
%Y A216349 First column gives: A216351.
%Y A216349 Last elements of rows give: A033917.
%Y A216349 A version with sorted row elements is: A216350.
%Y A216349 Rows sums give: A216281.
%Y A216349 Cf. A000081, A215703.
%K A216349 nonn,look,tabf
%O A216349 1,2
%A A216349 _Alois P. Heinz_, Sep 04 2012