A294747 Number of compositions (ordered partitions) of 1 into exactly n^2+1 powers of 1/(n+1).

1, 1, 10, 4245, 216456376, 2713420774885145, 14138484434475011392912026, 46050764886573707269872023694736134925, 131223281654667714701311635640432890136981994039662720, 435699237793484726791774188056400878106883117166142375354233228879800569

Offset: 0

Author

Alois P. Heinz, Nov 07 2017

Keywords

• nonn

Examples

			a(0) = 1: [1].
a(1) = 1: [1/2,1/2].
a(2) = 10 = binomial(5,2): [1/3,1/3,1/9,1/9,1/9], [1/3,1/9,1/3,1/9,1/9], [1/3,1/9,1/9,1/3,1/9], [1/3,1/9,1/9,1/9,1/3], [1/9,1/3,1/3,1/9,1/9], [1/9,1/3,1/9,1/3,1/9], [1/9,1/3,1/9,1/9,1/3], [1/9,1/9,1/3,1/3,1/9], [1/9,1/9,1/3,1/9,1/3], [1/9,1/9,1/9,1/3,1/3].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..26

Crossrefs

Main diagonal of A294746.

Cf. A002522.

Programs

Maple

b:= proc(n, r, p, k) option remember;
      `if`(n `if`(n=0, 1, b(n^2+1, 1, 0, n+1)):
seq(a(n), n=0..10);

Mathematica

b[n_, r_, p_, k_] := b[n, r, p, k] = If[n < r, 0, If[r == 0, If[n == 0, p!, 0], Sum[b[n - j, k*(r - j), p + j, k]/j!, {j, 0, Min[n, r]}]]];
a[n_] := If[n == 0, 1, b[n^2 + 1, 1, 0, n + 1]];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

Formula

a(n) = [x^((n+1)^n)] (Sum_{j=0..n^2+1} x^((n+1)^j))^(n^2+1) for n>0, a(0) = 1.

a(n) = A294746(n,n).

a(n) ~ exp(-1/12) * n^(n^2 - n/2 + 2) / (2*Pi)^((n-1)/2). - Vaclav Kotesovec, Sep 20 2019