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, 800846, 1018083, 1289282, 1626753, 2045379, 2563137, 3201664, 3986975, 4950255, 6128842

Offset

1,2

Comments

Partial sum operator applied to partition numbers 4 times.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10000

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

a(n) = Sum_{k=1..n} A000041(k)*A000217(n+1-k). - Andrew Howroyd, Oct 29 2025

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 *)

Prog

(PARI) seq(n)=Vec(sum(k=1, n, numbpart(k)*x^k, O(x*x^n))/(1-x)^3) \\ Andrew Howroyd, Oct 29 2025

Crossrefs

Partial sums of A085360.

Cf. A000041, A000217, A014160, A182738, A259279.

Sequence in context: A093802 A006008 A325952 * A046776 A360486 A144898

Adjacent sequences: A086713 A086714 A086715 * A086717 A086718 A086719

Keyword

nonn

Author

Jon Perry, Jul 29 2003

Extensions

a(31) onward from Andrew Howroyd, Oct 29 2025

Status

approved