A086716 Convolution of triangular numbers with partition numbers.

1, 5, 15, 36, 75, 143, 255, 433, 707, 1119, 1725, 2602, 3851, 5607, 8046, 11399, 15963, 22123, 30369, 41328, 55792, 74763, 99496, 131566, 172931, 226027, 293864, 380160, 489480, 627428

Offset: 1

Author

Jon Perry, Jul 29 2003

Keywords

• nonn

Comments

Partial sum operator applied to partition numbers 4 times.

Crossrefs

Cf. A000041, A000217, A182738, A259279.

Programs

Mathematica

s1=s2=s3=0;lst={};Do[AppendTo[lst,s3+=s2+=s1+=PartitionsP[n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 16 2009 *)
Table[Sum[PartitionsP[k]*(n-k+1)*(n-k+2)/2,{k,1,n}],{n,1,50}] (* Vaclav Kotesovec, Jun 23 2015 *)

Formula

a(n) = ((n+1)*(n+2)*(A000070(n)-1) - (2*n+3)*A182738(n) + A259279(n))/2. - Vaclav Kotesovec, Jun 23 2015

a(n) ~ 3*sqrt(n) * exp(Pi*sqrt(2*n/3)) / (sqrt(2)*Pi^3). - Vaclav Kotesovec, Jun 23 2015