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 A166941 Product plus sum of four consecutive nonnegative numbers. 6

%I

%S 6,34,134,378,862,1706,3054,5074,7958,11922,17206,24074,32814,43738,

%T 57182,73506,93094,116354,143718,175642,212606,255114,303694,358898,

%U 421302,491506,570134,657834,755278,863162,982206,1113154,1256774

%N Product plus sum of four consecutive nonnegative numbers.

%C a(n) = (n*...*(n+3))+(n+...+(n+3)), n >= 0.

%C All terms are even.

%C Binomial transform of 2*A167858.

%H Vincenzo Librandi, <a href="/A166941/b166941.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = n^4 + 6*n^3 + 11*n^2 + 10*n + 6. [From _Charles R Greathouse IV_, Nov 02 2009, corrected by Klaus Brockhaus, Nov 14 2009]

%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4)+24 for n > 3; a(0)=6, a(1)=34, a(2)=134, a(3)=378. [From Klaus Brockhaus, Nov 14 2009]

%F G.f.: 2*(3+2*x+12*x^2-6*x^3+x^4)/(1-x)^5. [From Klaus Brockhaus, Nov 14 2009]

%e a(0) = 0*1*2*3+0+1+2+3 = 0+6 = 6.

%e a(1) = 1*2*3*4+1+2+3+4 = 24+10 = 34.

%t lst={};Do[p=(n+3)*(n+2)*(n+1)*n+(n+3)+(n+2)+(n+1)+n;AppendTo[lst,p],{n,0,5!}];lst

%o (MAGMA) [ &*s + &+s where s is [n..n+3]: n in [0..32] ]; [From Klaus Brockhaus, Nov 14 2009]

%Y Cf. A001477 (nonnegative integers), A167858 (3,14,36,36,12,0,0,0,...), A028387 (n+(n+1)^2), A167875, A166942, A166943.

%K nonn,easy

%O 0,1

%A _Vladimir Joseph Stephan Orlovsky_, Oct 24 2009

%E Edited and offset corrected by _Klaus Brockhaus_, Nov 14 2009

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