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).

1, 2, 12, 9, 156, 100, 80, 56, 3160, 1880, 1180, 1420, 950, 1360, 890, 660, 480, 87990, 50496, 29682, 35382, 24042, 22008, 14928, 31968, 20268, 14988, 10848, 34974, 21474, 13314, 15114, 10974, 13014, 8874, 6534, 5094, 3218628, 1806476, 1021552, 588756, 1189132

Offset: 1

Author

Alois P. Heinz, Sep 04 2012

Keywords

• nonn
• look
• tabf

Comments

The ordering of the functions is the same as in A215703 and is defined by the algorithm below.

Examples

			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.
Triangle T(n,k) begins:
:     1;
:     2;
:    12,     9;
:   156,   100,    80,    56;
:  3160,  1880,  1180,  1420,   950,  1360,   890,   660,   480;
: 87990, 50496, 29682, 35382, 24042, 22008, 14928, 31968, 20268, ...
		

Links

• Alois P. Heinz, Rows n = 1..12, flattened

Crossrefs

First column gives: A216351.

Last elements of rows give: A033917.

A version with sorted row elements is: A216350.

Rows sums give: A216281.

Cf. A000081, A215703.

Programs

Maple

with(combinat):
F:= proc(n) F(n):= `if`(n<2, [x$n], map(h->x^h, g(n-1, n-1))) end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, [x^n],
     `if`(i<1, [], [seq(seq(seq(mul(F(i)[w[t]-t+1], t=1..j)*v,
      w=choose([$1..nops(F(i))+j-1], j)), v=g(n-i*j, i-1)), j=0..n/i)]))
    end:
T:= n-> map(f-> n!*coeff(series(subs(x=x+1, f), x, n+1), x, n), F(n))[]:
seq(T(n), n=1..7);