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A215834 Fourth derivative of f_n at x=1, where f_n is the n-th of all functions that are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways.

%I A215834 #12 Mar 06 2019 21:02:34
%S A215834 0,8,52,32,156,100,80,56,344,228,148,172,124,152,104,80,56,640,440,
%T A215834 300,324,252,220,172,268,196,148,124,248,176,128,128,104,104,80,56,56,
%U A215834 1068,760,536,372,560,464,396,324,292,244,196,444,348,276,252,316,244
%N A215834 Fourth derivative of f_n at x=1, where f_n is the n-th of all functions that are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways.
%C A215834 For the ordering of functions f_n see A215703.
%H A215834 Alois P. Heinz, <a href="/A215834/b215834.txt">Table of n, a(n) for n = 1..7813</a>
%p A215834 T:= proc(n) T(n):=`if`(n=1, [x], map(h-> x^h, g(n-1$2))) end:
%p A215834 g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(
%p A215834       seq(mul(T(i)[w[t]-t+1], t=1..j)*v, v=g(n-i*j, i-1)), w=
%p A215834       combinat[choose]([$1..nops(T(i))+j-1], j)), j=0..n/i)])
%p A215834     end:
%p A215834 f:= proc() local i, l; i, l:= 0, []; proc(n) while n>
%p A215834       nops(l) do i:= i+1; l:= [l[], T(i)[]] od; l[n] end
%p A215834     end():
%p A215834 a:= n-> 4!*coeff(series(subs(x=x+1, f(n)), x, 5), x, 4):
%p A215834 seq(a(n), n=1..100);
%Y A215834 Row n=4 of A215703.
%Y A215834 Number of distinct values of a(n) taken for functions with m x's: A199205.
%Y A215834 Cf. A000081, A087803.
%K A215834 nonn
%O A215834 1,2
%A A215834 _Alois P. Heinz_, Aug 24 2012