A209817 - OEIS

%I #14 Oct 28 2015 13:21:37

%S 0,1,5,19,54,141,331,733,1527,3060,5888,11004,19978,35452,61538,

%T 104875,175618,289656,470914,755880,1198693,1880246,2918919,4488553,

%U 6840398,10337947,15500575,23070000,34094908,50055877,73026093,105902689,152706404,219004225

%N Number of partitions of 3n in which every part is <n.

%H Alois P. Heinz, <a href="/A209817/b209817.txt">Table of n, a(n) for n = 1..1000</a>

%e The 5 partitions of 9 with parts <3 are as follows:

%e 2+2+2+2+1

%e 2+2+2+1+1+1

%e 2+2+1+1+1+1+1

%e 2+1+1+1+1+1+1+1

%e 1+1+1+1+1+1+1+1+1.

%p b:= proc(n, i) option remember; `if`(n=0, 1,

%p       `if`(i<1, 0, b(n, i-1) +`if`(i>n, 0, b(n-i, i))))

%p     end:

%p a:= n-> b(3*n, n-1):

%p seq(a(n), n=1..50);  # _Alois P. Heinz_, Jul 09 2012

%t f[n_] := Length[Select[IntegerPartitions[3 n], First[#] <= n - 1 &]]; Table[f[n], {n, 1, 25}] (* A209817 *)

%t 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-1]; Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Oct 28 2015, after _Alois P. Heinz_ *)

%Y Cf. A209818.

%K nonn

%O 1,3

%A _Clark Kimberling_, Mar 13 2012

%E More terms from _Alois P. Heinz_, Jul 09 2012