cp's OEIS Frontend

A026908 a(n) = A026907(2*n, n).

%I A026908 #17 Aug 26 2025 09:01:24
%S A026908 1,44,198,758,2824,10530,39516,149274,567120,2164898,8297206,31907214,
%T A026908 123053938,475753142,1843357752,7155936162,27826486272,108369397602,
%U A026908 422614721742,1650118887102,6450143419962,25238550057222
%N A026908 a(n) = A026907(2*n, n).
%H A026908 G. C. Greubel, <a href="/A026908/b026908.txt">Table of n, a(n) for n = 0..1000</a>
%F A026908 a(n) = 4^n*Gamma(n+1/2)*(38 + 99*n + 49*n^2)*Pi^(-1/2)/Gamma(n+3) - 18 (based on guessed recurrence) - _Mark van Hoeij_, Oct 30 2011
%F A026908 From _G. C. Greubel_, Aug 22 2025: (Start)
%F A026908 a(n) = binomial(2*n, n) + 3*binomial(2*n+4, n+2) - 18.
%F A026908 a(n) = (n+2)*A000108(n) + 3*(n+3)*A000108(n+2) - 18.
%F A026908 G.f.: 2*(7 + 45*x + 2*x^2 - (5 - 23*x)*sqrt(1-4*x))/((1-x)*sqrt(1-4*x)*(1 + sqrt(1 - 4*x))^2).
%F A026908 E.g.f.: (1/x)*exp(2*x)*( 25*x*BesselI(0, 2*x) - 6*(1-4*x)*BesselI(1, 2*x) ) - 18*exp(x). (End)
%t A026908 With[{b=Binomial}, Table[b[2*n,n] +3*b[2*n+4,n+2] -18, {n,0,40}]] (* _G. C. Greubel_, Aug 22 2025 *)
%o A026908 (Magma)
%o A026908 A026908:= func< n | (n+1)*Catalan(n) +3*(n+3)*Catalan(n+2) -18 >;
%o A026908 [A026908(n): n in [0..40]]; // _G. C. Greubel_, Aug 22 2025
%o A026908 (SageMath)
%o A026908 def A026908(n): return binomial(2*n,n) +3*binomial(2*n+4,n+2) -18
%o A026908 print([A026908(n) for n in range(41)]) # _G. C. Greubel_, Aug 22 2025
%Y A026908 Cf. A000108, A026907.
%K A026908 nonn,changed
%O A026908 0,2
%A A026908 _Clark Kimberling_