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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). 8
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 20 14:54 EST 2019. Contains 329337 sequences. (Running on oeis4.)