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 A360103 #13 Mar 12 2023 09:23:08
%S A360103 1,2,9,49,283,1715,10793,69906,463031,3122264,21363065,147951489,
%T A360103 1035173405,7306326465,51959150713,371950057003,2678083379707,
%U A360103 19381867703946,140915907625531,1028760981192771,7538511404971231,55427326349613665,408789584900354397
%N A360103 a(n) = Sum_{k=0..n} binomial(n+4*k,n-k) * Catalan(k).
%F A360103 G.f. A(x) satisfies A(x) = 1/(1-x) + x * A(x)^2 / (1-x)^4.
%F A360103 G.f.: (1/(1-x)) * c(x/(1-x)^5), where c(x) is the g.f. of A000108.
%F A360103 D-finite with recurrence (n+1)*a(n) +2*(-5*n+3)*a(n-1) +(19*n-47)*a(n-2) +20*(-n+4)*a(n-3) +5*(3*n-17)*a(n-4) +2*(-3*n+22)*a(n-5) +(n-9)*a(n-6)=0. - _R. J. Mathar_, Mar 12 2023
%p A360103 A360103 := proc(n)
%p A360103 add(binomial(n+4*k,n-k)*A000108(k),k=0..n) ;
%p A360103 end proc:
%p A360103 seq(A360103(n),n=0..40) ; # _R. J. Mathar_, Mar 12 2023
%o A360103 (PARI) a(n) = sum(k=0, n, binomial(n+4*k, n-k)*binomial(2*k, k)/(k+1));
%o A360103 (PARI) my(N=30, x='x+O('x^N)); Vec(2/((1-x)*(1+sqrt(1-4*x/(1-x)^5))))
%Y A360103 Partial sums of A360101.
%Y A360103 Cf. A000108, A360057.
%Y A360103 Cf. A006318, A007317, A162476, A360102.
%K A360103 nonn
%O A360103 0,2
%A A360103 _Seiichi Manyama_, Jan 25 2023