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

0,3

Links

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

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

Example

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

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 *)

Crossrefs

Main diagonal of A292622.

Cf. A292462, A292463, A292503.

Sequence in context: A369578 A210496 A067676 * A307465 A116703 A007443

Adjacent sequences: A292504 A292505 A292506 * A292508 A292509 A292510

Keyword

nonn

Author

Alois P. Heinz, Sep 17 2017

Status

approved