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 A026909 #11 Aug 26 2025 04:50:48
%S A026909 22,99,379,1412,5265,19758,74637,283560,1082449,4148603,15953607,
%T A026909 61526969,237876571,921678876,3577968081,13913243136,54184698801,
%U A026909 211307360871,825059443551,3225071709981,12619275028611,49423455006501,193732625020419,760001601263697,2983614441691035,11720995167614703,46074369462135607
%N A026909 a(n) = (1/2)*A026907(2*n, n).
%H A026909 G. C. Greubel, <a href="/A026909/b026909.txt">Table of n, a(n) for n = 1..1000</a>
%F A026909 From _G. C. Greubel_, Aug 22 2025: (Start)
%F A026909 a(n) = (binomial(2*n, n) + 3*binomial(2*n+4, n+2) - 18)/2.
%F A026909 a(n) = ((n+2)*A000108(n) + 3*(n+3)*A000108(n+2))/2 - 9.
%F A026909 G.f.: 2*(3 + 25*x - x^2 - (3 - 13*x + x^2)*sqrt(1-4*x))/((1-x)*sqrt(1-4*x)*(1 + sqrt(1 - 4*x))^2).
%F A026909 E.g.f.: (1/(2*x))*exp(2*x)*( 25*x*BesselI(0, 2*x) - 6*(1-4*x)*BesselI(1, 2*x) ) - 9*exp(x) - 1/2. (End)
%t A026909 With[{b=Binomial}, Table[(b[2*n,n] +3*b[2*n+4,n+2] -18)/2, {n,40}]] (* _G. C. Greubel_, Aug 22 2025 *)
%o A026909 (Magma)
%o A026909 A026909:= func< n | ((n+1)*Catalan(n) +3*(n+3)*Catalan(n+2))/2 -9 >;
%o A026909 [A026909(n): n in [1..40]]; // _G. C. Greubel_, Aug 22 2025
%o A026909 (SageMath)
%o A026909 def A026909(n): return (binomial(2*n,n) +3*binomial(2*n+4,n+2))//2 -9
%o A026909 print([A026909(n) for n in range(1,41)]) # _G. C. Greubel_, Aug 22 2025
%Y A026909 Cf. A000108, A026907, A026908.
%K A026909 nonn,changed
%O A026909 1,1
%A A026909 _Clark Kimberling_
%E A026909 More terms added by _G. C. Greubel_, Aug 22 2025