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 A360102 #22 Sep 02 2024 15:14:20
%S A360102 1,2,7,30,141,703,3655,19603,107679,602756,3426049,19721069,114728723,
%T A360102 673494466,3984493735,23732956453,142204128507,856560123504,
%U A360102 5183708936061,31502904805922,192180259402691,1176416604202925,7223943302003917,44486888142708088
%N A360102 a(n) = Sum_{k=0..n} binomial(n+2*k,n-k) * Catalan(k).
%F A360102 G.f. A(x) satisfies A(x) = 1/(1-x) + x * A(x)^2 / (1-x)^2.
%F A360102 G.f.: (1/(1-x)) * c(x/(1-x)^3), where c(x) is the g.f. of A000108.
%F A360102 D-finite with recurrence (n+1)*a(n) +4*(-2*n+1)*a(n-1) +10*(n-2)*a(n-2) +2*(-2*n+7)*a(n-3) +(n-5)*a(n-4)=0. - _R. J. Mathar_, Mar 12 2023
%p A360102 A360102 := proc(n)
%p A360102 add(binomial(n+2*k,n-k)*A000108(k),k=0..n) ;
%p A360102 end proc:
%p A360102 seq(A360102(n),n=0..70) ; # _R. J. Mathar_, Mar 12 2023
%o A360102 (PARI) a(n) = sum(k=0, n, binomial(n+2*k, n-k)*binomial(2*k, k)/(k+1));
%o A360102 (PARI) my(N=30, x='x+O('x^N)); Vec(2/((1-x)*(1+sqrt(1-4*x/(1-x)^3))))
%Y A360102 Partial sums of A360100.
%Y A360102 Partial sums are A258973.
%Y A360102 Cf. A000108, A162481.
%Y A360102 Cf. A006318, A007317, A162476, A360103.
%K A360102 nonn
%O A360102 0,2
%A A360102 _Seiichi Manyama_, Jan 25 2023