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

1, 2, 12, 112, 1390, 21324, 387674, 8126000, 192616470, 5089321300, 148225991386, 4716320842248, 162745503111542, 6053000082586940, 241386577491939450, 10274734610562571360, 464969951693639429398, 22292508702711459409956, 1128813253960656111451418, 60200897135221442194205240

Offset: 1

Author

Luc Rousseau, Jul 12 2024

Keywords

• nonn

Comments

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.

Examples

			a(1) = 1^1 = 1.
a(2) = 2^2 - 2^1*a(1) = 2.
a(3) = 3^3 - 3^2*a(1) - 3^1*a(2) = 12.
a(1) = + 1^1                  ( 0---1 )
     = 1.
a(2) = + 2^2                  ( 0-------2 )
       - 2^1 * 1^1            ( 0---1---2 )
     = 2.
a(3) = + 3^3                  ( 0-----------3 )
       - 3^2 * 1^1            ( 0---1-------3 )
       - 3^1 * 2^2            ( 0-------2---3 )
       + 3^1 * 2^1 * 1^1      ( 0---1---2---3 )
     = 12.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..386

Crossrefs

Cf. A374601.

Programs

Maple

a:= proc(n) option remember; `if`(n<1, 0,
       n^n-add(n^(n-i)*a(i), i=1..n-1))
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Jul 13 2024

Mathematica

a[n_] := a[n] = n^n - Sum[n^(n - i)*a[i], {i, 1, n - 1}]
a /@ Range[20]

PARI

a(n)=n^n-sum(i=1,n-1,n^(n-i)*a(i))

Formula

a(n) = n^n - Sum_{i=1..n-1} n^(n-i)*a(i).

a(n) = -Sum_{c composition of n} ((-1)^(#c) * Product_{k=1..#c} (n - (Sum_{i

a(n) = n * A374601(n).

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