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 A277220 #23 Sep 08 2022 08:46:17
%S A277220 0,1,3,11,43,180,790,3590,16745,79705,385615,1890747,9375216,46931897,
%T A277220 236873261,1204089630,6159064015,31678706490,163739008070,
%U A277220 850051218980,4430529313065,23175017046351,121617754070653,640122809255716,3378402106118508,17875011275340275
%N A277220 Exponential convolution of Fibonacci (A000045) and Catalan (A000108) numbers.
%H A277220 G. C. Greubel, <a href="/A277220/b277220.txt">Table of n, a(n) for n = 0..1000</a>
%F A277220 a(n) = Sum_{k=0..n} binomial(n,k) * A000045(k) * A000108(n-k).
%F A277220 a(n) = (phi^n * hypergeom([1/2, -n], [2], -4/phi) - (-phi)^(-n) * hypergeom([1/2, -n], [2], 4*phi))/sqrt(5), where phi = (1+sqrt(5))/2 = A001622.
%F A277220 Recurrence: 19*(n+1)*(n+2)*(11*n+13)*a(n) + 2*(55*n^3+208*n^2+311*n+230)*a(n+1) + 2*(55*n^3+373*n^2+674*n+206)*a(n+3) = (n+2)*(297*n^2+1022*n+617)*a(n+2) + (n+3)*(n+5)*(11*n+2)*a(n+4).
%F A277220 E.g.f.: 2*exp(5*x/2)*sinh(x*sqrt(5)/2)*(BesselI_0(2*x) - BesselI_1(2*x))/sqrt(5) (the product of e.g.f. for Fibonacci and Catalan numbers).
%F A277220 a(n) ~ (phi + 4)^(n + 3/2) / (8 * sqrt(5*Pi) * n^(3/2)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Mar 10 2018
%t A277220 Table[Sum[Binomial[n, k] Fibonacci[k] CatalanNumber[n - k], {k, 0, n}], {n, 0, 30}] (* or *)
%t A277220 Round@Table[(GoldenRatio^n Hypergeometric2F1[1/2, -n, 2, -4/GoldenRatio] - (-GoldenRatio)^(-n) Hypergeometric2F1[1/2, -n, 2, 4 GoldenRatio])/Sqrt[5], {n, 0, 30}] (* Round is equivalent to FullSimplify here, but is much faster *)
%o A277220 (PARI) for(n=0, 30, print1(sum(k=0,n, binomial(n,k)*fibonacci(k)* binomial(2*n-2*k,n-k)/(n-k+1)), ", ")) \\ _G. C. Greubel_, Oct 22 2018
%o A277220 (Magma) [(&+[Binomial(n,k)*Fibonacci(k)*Catalan(n-k): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Oct 22 2018
%Y A277220 Cf. A277251, A000045, A000108, A014330, A014334, A090826.
%K A277220 nonn
%O A277220 0,3
%A A277220 _Vladimir Reshetnikov_, Oct 06 2016