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 A027790 #29 Feb 22 2025 09:56:49
%S A027790 10,80,350,1120,2940,6720,13860,26400,47190,80080,130130,203840,
%T A027790 309400,456960,658920,930240,1288770,1755600,2355430,3116960,4073300,
%U A027790 5262400,6727500,8517600,10687950,13300560,16424730,20137600
%N A027790 a(n) = 10*(n+1)*binomial(n+3,5)/3.
%C A027790 Number of 9-subsequences of [ 1, n ] with just 3 contiguous pairs.
%H A027790 G. C. Greubel, <a href="/A027790/b027790.txt">Table of n, a(n) for n = 2..1000</a>
%H A027790 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A027790 G.f.: 10*(1+x)*x^2/(1-x)^7.
%F A027790 a(n) = binomial(n+1, 3)*binomial(n+3, 3) = A000292(n-1)*A000292(n+1). - _Zerinvary Lajos_, May 13 2005
%F A027790 a(n) = 10*A040977(n). - _R. J. Mathar_, May 22 2013
%F A027790 From _Amiram Eldar_, Jan 06 2021: (Start)
%F A027790 Sum_{n>=2} 1/a(n) = 3*Pi^2/2 - 235/16.
%F A027790 Sum_{n>=2} (-1)^n/a(n) = 3*Pi^2/4 - 117/16. (End)
%F A027790 E.g.f.: (1/36)*x^2*(180 + 300*x + 135*x^2 + 21*x^3 + x^4)*exp(x). - _G. C. Greubel_, Feb 21 2025
%t A027790 Table[10(n+1) Binomial[n+3,5]/3,{n,2,30}] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{10,80,350,1120,2940,6720,13860},30] (* _Harvey P. Dale_, Jan 15 2015 *)
%o A027790 (Magma)
%o A027790 A027790:= func< n | 10*(n+1)*Binomial(n+3,5)/3 >;
%o A027790 [A027790(n): n in [2..40]]; // _G. C. Greubel_, Feb 21 2025
%o A027790 (SageMath)
%o A027790 def A027790(n): return 10*(n+1)*binomial(n+3,5)//3
%o A027790 print([A027790(n) for n in range(2,41)]) # _G. C. Greubel_, Feb 21 2025
%Y A027790 Cf. A000292, A040977.
%K A027790 nonn,easy
%O A027790 2,1
%A A027790 thi ngoc dinh (via _R. K. Guy_)