A166942 One fifth of product plus sum of five consecutive nonnegative numbers.

2, 27, 148, 509, 1350, 3031, 6056, 11097, 19018, 30899, 48060, 72085, 104846, 148527, 205648, 279089, 372114, 488395, 632036, 807597, 1020118, 1275143, 1578744, 1937545, 2358746, 2850147, 3420172, 4077893, 4833054, 5696095

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, Oct 24 2009

Keywords

• nonn
• easy

Comments

a(n) = ((n*...*(n+4))+(n+...+(n+4)))/5, n >= 0.

Binomial transform of 2, 25, 96, 144, 96, 24, 0, 0, 0, 0, ....

Partial sums of A062938 where initial term 1 is replaced by 2.

Examples

			a(0) = (0*1*2*3*4 + 0 + 1 + 2 + 3 + 4)/5 = (0 + 10)/5 = 2.
a(1) = (1*2*3*4*5 + 1 + 2 + 3 + 4 + 5)/5 = (120 + 15)/5 = 27.
		

Links

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

Crossrefs

Cf. A001477 (nonnegative integers), A062938 (squares of the form n(n+1)(n+2)(n+3)+1), A028387 (n+(n+1)^2), A167875, A166941, A166943.

Programs

Magma

[ (&*s + &+s)/5 where s is [n..n+4]: n in [0..29] ]; // Klaus Brockhaus, Nov 14 2009

Mathematica

Table[((n+4)*(n+3)*(n+2)*(n+1)*n+(n+4)+(n+3)+(n+2)+(n+1)+n)/5, {n,0,100}]
(Total[#]+Times@@#)/5&/@Partition[Range[0,100],5,1]  (* Harvey P. Dale, Mar 05 2011 *)

Formula

a(n) = (n^5 + 10n^4 + 35n^3 + 50n^2 + 29n + 10)/5. - Charles R Greathouse IV, Nov 02 2009

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + 24 for n > 4; a(0)=2, a(1)=27, a(2)=148, a(3)=509, a(4)=1350. - Klaus Brockhaus, Nov 14 2009

G.f.: (2+15*x+16*x^2-14*x^3+6*x^4-x^5)/(1-x)^6. - Klaus Brockhaus, Nov 14 2009

Extensions

Edited and offset corrected by Klaus Brockhaus, Nov 14 2009