cp's OEIS Frontend

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

%I A209818 #31 May 07 2018 10:12:11
%S A209818 1,4,12,34,84,199,436,919,1845,3590,6751,12384,22142,38797,66634,
%T A209818 112540,187013,306421,495332,791131,1249153,1951915,3019969,4630063,
%U A209818 7037286,10610240,15874998,23582081,34791668,50999319,74297620,107608848,154986104,222037997
%N A209818 Number of partitions of 3n in which every part is <=n.
%C A209818 Also, the number of partitions of 4n into n parts. - _Seiichi Manyama_, May 07 2018
%H A209818 Alois P. Heinz, <a href="/A209818/b209818.txt">Table of n, a(n) for n = 1..1000</a>
%F A209818 For n>3, a(n) = A000041(3*n) - A000070(2*n-1) + A000097(n-3). - _Vaclav Kotesovec_, May 25 2015
%F A209818 a(n) ~ exp(Pi*sqrt(2*n))/(12*sqrt(3)*n). - _Vaclav Kotesovec_, May 25 2015
%e A209818 The 4 partitions of 6 with parts <3:
%e A209818 2+2+2, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1.
%e A209818 From _Seiichi Manyama_, May 07 2018: (Start)
%e A209818 n | Partitions of 4n into n parts
%e A209818 --+-------------------------------------------
%e A209818 1 | 4;
%e A209818 2 | 7+1, 6+2, 5+3, 4+4;
%e A209818 3 | 10+1+1, 9+2+1, 8+3+1, 8+2+2, 7+4+1, 7+3+2,
%e A209818   |  6+5+1, 6+4+2, 6+3+3, 5+5+2, 5+4+3, 4+4+4; (End)
%p A209818 b:= proc(n, i) option remember;
%p A209818       `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
%p A209818     end:
%p A209818 a:= n-> b(3*n, n):
%p A209818 seq(a(n), n=1..50);  # _Alois P. Heinz_, Jul 09 2012
%t A209818 f[n_] := Length[Select[IntegerPartitions[3n], First[#] <= n &]]; Table[f[n], {n, 1, 25}] (* A209818 *)
%t A209818 Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,3*n}],{n,1,20}] (* _Vaclav Kotesovec_, May 25 2015 *)
%t A209818 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_ *)
%t A209818 Table[Length@IntegerPartitions[4n, {n}], {n, 25}] (* _Vladimir Reshetnikov_, Jul 24 2016 *)
%Y A209818 Cf. A209817.
%K A209818 nonn
%O A209818 1,2
%A A209818 _Clark Kimberling_, Mar 13 2012
%E A209818 More terms from _Alois P. Heinz_, Jul 09 2012