A166943 One third of product plus sum of six consecutive nonnegative numbers.

5, 247, 1689, 6731, 20173, 50415, 110897, 221779, 411861, 720743, 1201225, 1921947, 2970269, 4455391, 6511713, 9302435, 13023397, 17907159, 24227321, 32303083, 42504045, 55255247, 71042449, 90417651, 114004853, 142506055

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, Oct 24 2009

Keywords

• nonn
• easy

Comments

a(n) = ((n*...*(n+5))+(n+...+(n+5)))/3, n >= 0.

Binomial transform of 5, 242, 1200, 2400, 2400, 1200, 240, 0, 0, 0, 0, ....

Examples

			a(0) = (0*1*2*3*4*5+0+1+2+3+4+5)/3 = (0+15)/3 = 5.
a(1) = (1*2*3*4*5*6+1+2+3+4+5+6)/3 = (720+21)/3 = 247.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A001477 (nonnegative integers), A028387 (n+(n+1)^2), A167875, A166941, A166942.

Programs

Magma

[ (&*s + &+s)/3 where s is [n..n+5]: n in [0..25] ]; // Klaus Brockhaus, Nov 14 2009

Mathematica

lst={};Do[p=(n+5)*(n+4)*(n+3)*(n+2)*(n+1)*n+(n+5)+(n+4)+(n+3)+(n+2)+(n+1)+n;AppendTo[lst,p/3],{n,0,5!}];lst
(Plus@@#+Times@@#)/3&/@Partition[Range[0,30],6,1] (* Harvey P. Dale, Nov 10 2009 *)

Formula

a(n) = (n^6 + 15n^5 + 85n^4 + 225n^3 + 274n^2 + 126n + 15)/3. - Charles R Greathouse IV, Nov 04 2009

a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6)+240 for n > 5; a(0)=5, a(1)=247, a(2)=1689, a(3)=6731, a(4)=20173, a(5)=50415. - Klaus Brockhaus, Nov 14 2009

G.f.: (5+212*x+65*x^2-80*x^3+55*x^4-20*x^5+3*x^6)/(1-x)^7. - Klaus Brockhaus, Nov 14 2009

Extensions

Edited and offset corrected by Klaus Brockhaus, Nov 14 2009