A292507 Number of partitions of n with up to n distinct kinds of 1.

1, 1, 2, 5, 13, 33, 82, 201, 488, 1176, 2817, 6714, 15931, 37647, 88628, 207914, 486158, 1133304, 2634339, 6106953, 14121157, 32573842, 74968044, 172164086, 394561089, 902471184, 2060338222, 4695324425, 10681885697, 24261437446, 55017434305, 124573678280

Offset: 0

Author

Alois P. Heinz, Sep 17 2017

Keywords

• nonn

Examples

			a(3) = 5: 3, 21a, 21b, 21c, 1a1b1c.
a(4) = 13: 4, 31a, 31b, 31c, 31d, 22, 21a1b, 21a1c, 21a1d, 21b1c, 21b1d, 21c1d, 1a1b1c1d.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1382 (terms 0..1000 from Alois P. Heinz)

Crossrefs

Main diagonal of A292622.

Cf. A292462, A292463, A292503.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or i=1,
      binomial(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..35);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, Binomial[k, n], If[i > n, 0, b[n - i, i, k]] + b[n, i - 1, k]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 20 2018, translated from Maple *)

Formula

Conjecture: log(a(n)) ~ log(2)*n + Pi*sqrt(n/3) - 3*log(n)/2. - Vaclav Kotesovec, May 11 2019

a(n) = [x^n] (1 + x)^n * Product_{k>=2} 1 / (1 - x^k). - Ilya Gutkovskiy, Apr 24 2021