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 A006637 M5005 #22 Sep 03 2025 10:50:51
%S A006637 16,96,344,952,2241,4712,9108,16488,28314,46552,73788,113360,169507,
%T A006637 247536,354008,496944,686052,932976,1251568,1658184,2172005,2815384,
%U A006637 3614220,4598360,5802030,7264296,9029556,11148064,13676487,16678496,20225392,24396768,29281208
%N A006637 Expansion of (2 - x)^4/(1 - x)^8.
%C A006637 Former name: From generalized Catalan numbers. - _G. C. Greubel_, Sep 03 2025
%D A006637 H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
%D A006637 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006637 G. C. Greubel, <a href="/A006637/b006637.txt">Table of n, a(n) for n = 0..1000</a>
%H A006637 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F A006637 G.f.: (2-x)^4/(1-x)^8. - _Sean A. Irvine_, May 31 2017
%F A006637 a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - _Wesley Ivan Hurt_, Jun 18 2022
%F A006637 From _G. C. Greubel_, Sep 03 2025: (Start)
%F A006637 a(n) = Sum_{k=0..4} binomial(4, k)*binomial(n+k+3, k+3).
%F A006637 a(n) = (1/7!)*(n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2 + 31*n + 192).
%F A006637 E.g.f.: (1/7!)*(80640 + 403200*x + 423360*x^2 + 161280*x^3 + 27090*x^4 + 2142*x^5 + 77*x^6 + x^7)*exp(x). (End)
%t A006637 Table[(n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2+31*n+192)/7!, {n,0,40}] (* _G. C. Greubel_, Sep 03 2025 *)
%o A006637 (Magma)
%o A006637 A006637:= func< n | (n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2+31*n+192)/5040 >;
%o A006637 [A006637(n): n in [0..40]]; // _G. C. Greubel_, Sep 03 2025
%o A006637 (SageMath)
%o A006637 def A006637(n): return (n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2+31*n+192)//5040
%o A006637 print([A006637(n) for n in range(41)]) # _G. C. Greubel_, Sep 03 2025
%Y A006637 Cf. A006633, A006634, A006635, A006636, A181289.
%K A006637 nonn,easy,changed
%O A006637 0,1
%A A006637 _Simon Plouffe_
%E A006637 a(6) and a(8) corrected and more terms from _Sean A. Irvine_, May 31 2017
%E A006637 New name by _G. C. Greubel_, Sep 03 2025