A332720 Index position of {3}^n within the list of partitions of 3n in canonical ordering.

1, 1, 5, 19, 59, 150, 349, 745, 1515, 2936, 5514, 10036, 17851, 31039, 53006, 88943, 147057, 239701, 385885, 613855, 966137, 1505137, 2323124, 3553914, 5392315, 8117758, 12131618, 18003740, 26543030, 38886999, 56633453, 82009410, 118113488, 169229009, 241264461

Offset

0,3

Comments

The canonical ordering of partitions is described in A080577.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..4000

Wikipedia, Integer Partition

Formula

a(n) ~ exp(Pi*sqrt(2*n)) / (4*3^(3/2)*n). - Vaclav Kotesovec, Feb 28 2020

Example

a(2) = 5, because 33 has position 5 within the list of partitions of 6 in canonical ordering: 6, 51, 42, 411, 33, 321, 3111, 222, ... .

Maple

b:= proc(n) option remember;
     `if`(n=0, 1, b(n-1)+g(3*n, 2))
    end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
     `if`(i<1, 0, g(n-i, min(n-i, i))+g(n, i-1)))
    end:
a:= n-> g(3*n$2)-b(n)+1:
seq(a(n), n=0..35);

Mathematica

b[n_] := b[n] = If[n == 0, 1, b[n - 1] + g[3n, 2]];
g[n_, i_] := g[n, i] = If[n == 0 || i == 1, 1, If[i < 1, 0, g[n - i, Min[n - i, i]] + g[n, i - 1]]];
a[n_] := g[3n, 3n] - b[n] + 1;
a /@ Range[0, 35] (* Jean-François Alcover, Jan 06 2021, after Alois P. Heinz *)

Crossrefs

Cf. A000041, A080577, A322761, A330661, A332706, A332719.

Sequence in context: A029861 A224034 A107179 * A092442 A341711 A328543

Adjacent sequences: A332717 A332718 A332719 * A332721 A332722 A332723

Keyword

nonn

Author

Alois P. Heinz, Feb 20 2020

Status

approved