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