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 A027173 #27 Sep 05 2025 11:16:52
%S A027173 5,30,123,472,1790,6794,25879,98952,379674,1461248,5638930,21811540,
%T A027173 84542016,328287506,1276868111,4973645576,19398954626,75753497816,
%U A027173 296142817406,1158865623236,4539024407576,17793358627976,69805770185498,274055019608372,1076651019788200
%N A027173 a(n) = A027170(2n, n-1).
%H A027173 Stefano Spezia, <a href="/A027173/b027173.txt">Table of n, a(n) for n = 1..1600</a>
%F A027173 From _Stefano Spezia_, Sep 02 2025: (Start)
%F A027173 a(n) = binomial(2*n,n-1) + 2*binomial(2*(1+n),n) - 4.
%F A027173 G.f.: (-2 + 2*sqrt(1-4*x) + 5*x - sqrt(1-4*x)*x - x^2 + 3*sqrt(1-4*x)*x^2 - 2*x^3 + 4*sqrt(1-4*x)*x^3)/(2*sqrt(1-4*x)*(-1 + x)*x^2).
%F A027173 E.g.f.: 2 - 4*exp(x) + 4*exp(2*x)*BesselI(0, 2*x) + (exp(2*x)*(5*x - 2)*BesselI(1, 2*x))/x. (End)
%t A027173 a[n_]:=Binomial[2 n,-1+n]+2 Binomial[2+2 n,n]-4; Array[a,30] (* _Stefano Spezia_, Sep 02 2025 *)
%t A027173 CoefficientList[Series[(-2 + 2*Sqrt[1-4*x] + 5*x - Sqrt[1-4*x]*x - x^2 + 3*Sqrt[1-4*x]*x^2 - 2*x^3 + 4*Sqrt[1-4*x]*x^3)/(2*Sqrt[1-4*x]*(-1 + x)*x^2),{x,0,25}],x] (* _Stefano Spezia_, Sep 02 2025 *)
%t A027173 CoefficientList[Series[2 - 4*Exp[x] + 4*Exp[2*x]*BesselI[0, 2*x] + (Exp[2*x]*(5*x - 2)*BesselI[1, 2*x])/x,{x,0,25}],x]*Range[0,25]! (* _Stefano Spezia_, Sep 02 2025 *)
%o A027173 (PARI) my(x='x+O('x^40)); Vec(serlaplace(2 - 4*exp(x) + 4*exp(2*x)*besseli(0, 2*x) + (exp(2*x)*(5*x - 2)*besseli(1, 2*x)))) \\ _Michel Marcus_, Sep 04 2025
%Y A027173 Cf. A027170.
%K A027173 nonn,changed
%O A027173 1,1
%A A027173 _Clark Kimberling_
%E A027173 a(24)-a(25) from _Stefano Spezia_, Sep 02 2025