%I #24 Jul 31 2024 09:09:20
%S 1,2,12,112,1390,21324,387674,8126000,192616470,5089321300,
%T 148225991386,4716320842248,162745503111542,6053000082586940,
%U 241386577491939450,10274734610562571360,464969951693639429398,22292508702711459409956,1128813253960656111451418,60200897135221442194205240
%N Defined by: Sum_{i=1..n} a(i) / n^i = 1, n >= 1.
%C Constant terms of the following polynomials: P(0,x) = -1 and, for n>0, P(n,x) = x*P(n-1,x) + a(n), a(n) chosen such that P(n,n)=0.
%H Seiichi Manyama, <a href="/A374562/b374562.txt">Table of n, a(n) for n = 1..386</a>
%F a(n) = n^n - Sum_{i=1..n-1} n^(n-i)*a(i).
%F a(n) = -Sum_{c composition of n} ((-1)^(#c) * Product_{k=1..#c} (n - (Sum_{i<k} c_i))^c_k).
%F a(n) = n * A374601(n).
%e a(1) = 1^1 = 1.
%e a(2) = 2^2 - 2^1*a(1) = 2.
%e a(3) = 3^3 - 3^2*a(1) - 3^1*a(2) = 12.
%e a(1) = + 1^1 ( 0---1 )
%e = 1.
%e a(2) = + 2^2 ( 0-------2 )
%e - 2^1 * 1^1 ( 0---1---2 )
%e = 2.
%e a(3) = + 3^3 ( 0-----------3 )
%e - 3^2 * 1^1 ( 0---1-------3 )
%e - 3^1 * 2^2 ( 0-------2---3 )
%e + 3^1 * 2^1 * 1^1 ( 0---1---2---3 )
%e = 12.
%p a:= proc(n) option remember; `if`(n<1, 0,
%p n^n-add(n^(n-i)*a(i), i=1..n-1))
%p end:
%p seq(a(n), n=1..20); # _Alois P. Heinz_, Jul 13 2024
%t a[n_] := a[n] = n^n - Sum[n^(n - i)*a[i], {i, 1, n - 1}]
%t a /@ Range[20]
%o (PARI) a(n)=n^n-sum(i=1,n-1,n^(n-i)*a(i))
%Y Cf. A374601.
%K nonn
%O 1,2
%A _Luc Rousseau_, Jul 12 2024