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A374562 Defined by: Sum_{i=1..n} a(i) / n^i = 1, n >= 1.

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