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).

1, 2, 9, 12, 56, 80, 100, 156, 480, 660, 890, 950, 1180, 1360, 1420, 1880, 3160, 5094, 6534, 8874, 10848, 10974, 13014, 13314, 14928, 14988, 15114, 20268, 21474, 22008, 24042, 29682, 31968, 34974, 35382, 50496, 87990, 65534, 78134, 102494, 131684, 141974

Offset

1,2

Links

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

Example

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].
Triangle T(n,k) begins:
:    1;
:    2;
:    9,   12;
:   56,   80,  100,   156;
:  480,  660,  890,   950,  1180,  1360,  1420,  1880,  3160;
: 5094, 6534, 8874, 10848, 10974, 13014, 13314, 14928, 14988, 15114, ...

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-> sort(map(f-> n!*coeff(series(subs(x=x+1, f)
                 , x, n+1), x, n), F(n)))[]:
seq(T(n), n=1..7);

Crossrefs

First column gives: A033917.

Last elements of rows give: A216351.

A version with different ordering of row elements is: A216349.

Rows sums give: A216281.

Cf. A000081, A215703.

Sequence in context: A076505 A218073 A129345 * A125019 A259984 A225548

Adjacent sequences: A216347 A216348 A216349 * A216351 A216352 A216353

Keyword

nonn,tabf

Author

Alois P. Heinz, Sep 04 2012

Status

approved