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Convolution of triangular numbers with partition numbers.
2

%I #16 Jun 23 2015 07:30:35

%S 1,5,15,36,75,143,255,433,707,1119,1725,2602,3851,5607,8046,11399,

%T 15963,22123,30369,41328,55792,74763,99496,131566,172931,226027,

%U 293864,380160,489480,627428

%N Convolution of triangular numbers with partition numbers.

%C Partial sum operator applied to partition numbers 4 times.

%F a(n) = ((n+1)*(n+2)*(A000070(n)-1) - (2*n+3)*A182738(n) + A259279(n))/2. - _Vaclav Kotesovec_, Jun 23 2015

%F a(n) ~ 3*sqrt(n) * exp(Pi*sqrt(2*n/3)) / (sqrt(2)*Pi^3). - _Vaclav Kotesovec_, Jun 23 2015

%t s1=s2=s3=0;lst={};Do[AppendTo[lst,s3+=s2+=s1+=PartitionsP[n]],{n,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Jul 16 2009 *)

%t Table[Sum[PartitionsP[k]*(n-k+1)*(n-k+2)/2,{k,1,n}],{n,1,50}] (* _Vaclav Kotesovec_, Jun 23 2015 *)

%Y Cf. A000041, A000217, A182738, A259279.

%K nonn

%O 1,2

%A _Jon Perry_, Jul 29 2003