A222731 - OEIS

%I #10 May 29 2018 09:20:24

%S 1,0,1,3,4,4,11,13,21,30,44,59,92,115,165,225,305,394,546,700,931,

%T 1204,1572,2005,2613,3290,4218,5328,6745,8423,10630,13193,16475,20386,

%U 25269,31072,38346,46882,57478,70066,85415,103582,125794,151916,183576,220962

%N Total sum of parts of multiplicity 3 in all partitions of n.

%H Alois P. Heinz, <a href="/A222731/b222731.txt">Table of n, a(n) for n = 3..1000</a>

%F G.f.: (x^3/(1-x^3)^2-x^4/(1-x^4)^2)/Product_{i>=1}(1-x^i).

%F a(n) ~ 7 * sqrt(3) * exp(Pi*sqrt(2*n/3)) / (288 * Pi^2). - _Vaclav Kotesovec_, May 29 2018

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

%p       add((l->`if`(m=3, l+[0, l[1]*p], l))(b(n-p*m, p-1)), m=0..n/p)))

%p     end:

%p a:= n-> b(n, n)[2]:

%p seq(a(n), n=3..50);

%t b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[5]]; Table[a[n], {n, 3, 50}] (* _Jean-François Alcover_, Jan 24 2014, after _Alois P. Heinz_ *)

%Y Column k=3 of A222730.

%K nonn

%O 3,4

%A _Alois P. Heinz_, Mar 03 2013