A292462 Number of partitions of n with n sorts of part 1.

1, 1, 5, 31, 278, 3287, 48256, 843567, 17081639, 392869430, 10112244792, 287927207846, 8984122319997, 304828239096197, 11173376516829974, 439988449921648076, 18523908107054523591, 830292183207722271065, 39475390430795389762048, 1984220622132901208082220

Offset: 0

Author

Alois P. Heinz, Sep 16 2017

Keywords

• nonn

Examples

			a(2) = 5: 2, 1a1a, 1a1b, 1b1a, 1b1b.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..386

Crossrefs

Cf. A002865, A246935, A292463, A292503, A292507, A292567.

Main diagonal of A292741.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^n,
      `if`(i>n, 0, b(n-i, i, k))+b(n, i-1, k))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..23);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, k^n, If[i > n, 0, b[n - i, i, k]] + b[n, i - 1, k]];
a[0] = 1; a[n_] := b[n, n, n];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, May 19 2018, translated from Maple *)

Formula

a(n) = [x^n] 1/(1-n*x) * Product_{j=2..n} 1/(1-x^j).

a(n) ~ n^n * (1 + 1/n^2 + 1/n^3 + 2/n^4 + 2/n^5 + 4/n^6 + 4/n^7 + 7/n^8 + 8/n^9 + 12/n^10), for coefficients see A002865. - Vaclav Kotesovec, Sep 19 2017

a(n) = Sum_{j=0..n} A002865(j) * n^(n-j). - Alois P. Heinz, Sep 22 2017