A061316 a(n) = n*(n+1)*(n^2 + n + 4)/4.

0, 3, 15, 48, 120, 255, 483, 840, 1368, 2115, 3135, 4488, 6240, 8463, 11235, 14640, 18768, 23715, 29583, 36480, 44520, 53823, 64515, 76728, 90600, 106275, 123903, 143640, 165648, 190095, 217155, 247008, 279840, 315843, 355215, 398160, 444888

Offset: 0

Author

Henry Bottomley, Apr 24 2001

Keywords

• nonn
• easy

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000217, A005563, A006007, A061314.

Programs

Maple

a:=n->sum((n+j^3),j=0..n): seq(a(n),n=0..36); # Zerinvary Lajos, Jul 27 2006
with(combinat):a:=n->sum(fibonacci(4,i), i=0..n): seq(a(n), n=0..36); # Zerinvary Lajos, Mar 20 2008

Mathematica

s=0;lst={};Do[s+=n^3+n*2;AppendTo[lst,s],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 04 2009 *)
Table[n(n+1)(n^2+n+4)/4,{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1}, {0,3,15,48,120},40] (* Harvey P. Dale, May 03 2011 *)

PARI

a(n) = n*(n + 1)*(n^2 + n + 4)/4 \\ Harry J. Smith, Jul 21 2009

Formula

a(n) = n*(n+1)*(n^2 + n + 4)/4.

a(n) = A005563(A000217(n)) = 3*A006007(n) = A061314(n, 2).

a(0)=0, a(1)=3, a(2)=15, a(3)=48, a(4)=120, a(n) = 5a(n-1) - 10a(n-2) + 10a(n-3) - 5a(n-4) + a(n-5).

G.f.: (-3 (x + x^3))/(-1 + x)^5. - Harvey P. Dale, May 03 2011

Sum_{n>=1} 1/a(n) = 5/4 - tanh(sqrt(15)*Pi/2)*Pi/sqrt(15). - Amiram Eldar, Aug 20 2022

E.g.f.: exp(x)*x*(12 + 18*x + 8*x^2 + x^3)/4. - Stefano Spezia, Aug 31 2023