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%I #20 Jan 06 2020 09:12:29
%S 1,1,6,42,420,5155,77658,1376340,28133616,651317463,16846515510,
%T 481472570920,15067838554860,512473599799551,18821719654854998,
%U 742395982266536520,31299550394528466960,1404629090174946183156,66851805805525048040334,3363381327122496643643628
%N a(n) is the n-th term of the inverse Euler transform of j-> n^(j-1).
%H Alois P. Heinz, <a href="/A306173/b306173.txt">Table of n, a(n) for n = 1..200</a>
%F a(n) ~ (1 - exp(-1)) * n^(n-1). - _Vaclav Kotesovec_, Oct 08 2019
%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(binomial(g(i, k)+j-1, j)*b(n-i*j, i-1,k), j=0..n/i)))
%p end:
%p g:= proc(n, k) option remember; k^(n-1)-b(n, n-1, k) end:
%p a:= n-> g(n$2):
%p seq(a(n), n=1..21);
%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[g[i, k] + j - 1, j]*b[n - i*j, i - 1, k], {j, 0, n/i}]]];
%t g[n_, k_] := g[n, k] = k^(n - 1) - b[n, n - 1, k];
%t a[n_] := g[n, n];
%t a /@ Range[21] (* _Jean-François Alcover_, Jan 06 2020, after _Alois P. Heinz_ *)
%Y Main diagonal of A065177.
%Y Cf. A075147, A316073.
%K nonn
%O 1,3
%A _Alois P. Heinz_, Jun 23 2018
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