This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A027659 #23 Nov 22 2024 07:57:37
%S A027659 4,18,52,121,246,455,784,1278,1992,2992,4356,6175,8554,11613,15488,
%T A027659 20332,26316,33630,42484,53109,65758,80707,98256,118730,142480,169884,
%U A027659 201348,237307,278226,324601,376960,435864,501908,575722,657972,749361,850630,962559
%N A027659 a(n) = binomial(n+2,2) + binomial(n+3,3) + binomial(n+4,4) + binomial(n+5,5).
%H A027659 Harvey P. Dale, <a href="/A027659/b027659.txt">Table of n, a(n) for n = 0..1000</a>
%H A027659 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A027659 a(n) = A035343(n+2, 5), n >= 0 (sixth column of quintinomial coefficients).
%F A027659 a(n) = A062750(n+2, 5), n >= 0 (sixth column).
%F A027659 G.f.: (x^2)*(2-x)*(2 - 2*x + x^2)/(1-x)^6. (For numerator polynomial see N5(5, x) = 4 - 6*x + 4*x^2 - x^3 from A063422.)
%F A027659 a(n) = binomial(n+6, 5) - binomial(n+2, 1). - _Zerinvary Lajos_, May 08 2006
%F A027659 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), with a(0)=4, a(1)=18, a(2)=52, a(3)=121, a(4)=246, a(5)=455. - _Harvey P. Dale_, Aug 18 2012
%F A027659 From _G. C. Greubel_, Aug 01 2022: (Start)
%F A027659 a(n) = Sum_{j=0..3} binomial(n+j+2, j+2).
%F A027659 E.g.f.: (1/120)*(480 +1680*x +1200*x^2 +300*x^3 +30*x^4 +x^5)*exp(x). (End)
%p A027659 seq(1/120*(n+8)*(n+2)*(n+1)*(n^2+9*n+30), n=0..40);
%t A027659 Table[Sum[Binomial[n+i,i],{i,2,5}],{n,0,30}] (* or *) LinearRecurrence[ {6,-15,20, -15,6,-1}, {4,18,52,121,246,455},30] (* _Harvey P. Dale_, Aug 18 2012 *)
%t A027659 Sum[(-1)^j*Binomial[4*j-2 + Range[0, 60], 4*j-3], {j,2}] (* _G. C. Greubel_, Aug 01 2022 *)
%o A027659 (PARI) a(n)=(n+8)*(n+2)*(n+1)*(n^2+9*n+30)/120 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A027659 (Magma) [Binomial(n+6, 5) -(n+2): n in [0..60]]; // _G. C. Greubel_, Aug 01 2022
%o A027659 (SageMath) [binomial(n+6, 5) -(n+2) for n in (0..60)] # _G. C. Greubel_, Aug 01 2022
%Y A027659 Cf. A035343, A062750, A063422.
%Y A027659 Partial sums of A063258.
%K A027659 nonn,easy
%O A027659 0,1
%A A027659 _N. J. A. Sloane_