A209818 Number of partitions of 3n in which every part is <=n.

1, 4, 12, 34, 84, 199, 436, 919, 1845, 3590, 6751, 12384, 22142, 38797, 66634, 112540, 187013, 306421, 495332, 791131, 1249153, 1951915, 3019969, 4630063, 7037286, 10610240, 15874998, 23582081, 34791668, 50999319, 74297620, 107608848, 154986104, 222037997

Offset

1,2

Comments

Also, the number of partitions of 4n into n parts. - Seiichi Manyama, May 07 2018

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

For n>3, a(n) = A000041(3*n) - A000070(2*n-1) + A000097(n-3). - Vaclav Kotesovec, May 25 2015

a(n) ~ exp(Pi*sqrt(2*n))/(12*sqrt(3)*n). - Vaclav Kotesovec, May 25 2015

Example

The 4 partitions of 6 with parts <3:
2+2+2, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1.
From Seiichi Manyama, May 07 2018: (Start)
n | Partitions of 4n into n parts
--+-------------------------------------------
1 | 4;
2 | 7+1, 6+2, 5+3, 4+4;
3 | 10+1+1, 9+2+1, 8+3+1, 8+2+2, 7+4+1, 7+3+2,
  |  6+5+1, 6+4+2, 6+3+3, 5+5+2, 5+4+3, 4+4+4; (End)

Maple

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> b(3*n, n):
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 09 2012

Mathematica

f[n_] := Length[Select[IntegerPartitions[3n], First[#] <= n &]]; Table[f[n], {n, 1, 25}] (* A209818 *)
Table[SeriesCoefficient[Product[1/(1-x^k), {k, 1, n}], {x, 0, 3*n}], {n, 1, 20}] (* Vaclav Kotesovec, May 25 2015 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[3*n, n]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 28 2015, after Alois P. Heinz *)
Table[Length@IntegerPartitions[4n, {n}], {n, 25}] (* Vladimir Reshetnikov, Jul 24 2016 *)

Crossrefs

Cf. A209817.

Sequence in context: A331834 A135373 A338695 * A094893 A036880 A349973

Adjacent sequences: A209815 A209816 A209817 * A209819 A209820 A209821

Keyword

nonn

Author

Clark Kimberling, Mar 13 2012

Extensions

More terms from Alois P. Heinz, Jul 09 2012

Status

approved