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A360143 a(n) = Sum_{k=0..n} binomial(2*n+2*k,n-k).

%I A360143 #23 Jul 23 2025 13:18:41
%S A360143 1,3,13,59,271,1250,5775,26696,123423,570576,2637306,12187755,
%T A360143 56312089,260134905,1201493926,5548533913,25619837773,118283258215,
%U A360143 546041467522,2520515546083,11633752319476,53693477980816,247798435809211,1143547904185879,5277058908767419
%N A360143 a(n) = Sum_{k=0..n} binomial(2*n+2*k,n-k).
%H A360143 Seiichi Manyama, <a href="/A360143/b360143.txt">Table of n, a(n) for n = 0..1000</a>
%F A360143 G.f.: 1 / ( sqrt(1-4*x) * (1 - x * c(x)^4) ), where c(x) is the g.f. of A000108.
%F A360143 D-finite with recurrence +n*(n-7)*a(n) -(7*n-4)*(n-7)*a(n-1) +4*(n^2-13*n+17)*a(n-2) +(35*n^2-217*n+304)*a(n-3) -2*(n-2)*(7*n-29)*a(n-4) +4*(n-2)*(2*n-9)*a(n-5)=0. - _R. J. Mathar_, Mar 12 2023
%F A360143 a(n) = binomial(2*n, n)*hypergeom([1, -n, 1/2+n, 1+n], [(1+n)/3, (2+n)/3, 1+n/3], -4/27). - _Stefano Spezia_, Jun 17 2025
%p A360143 A360143 := proc(n)
%p A360143     add(binomial(2*n+2*k,n-k),k=0..n) ;
%p A360143 end proc:
%p A360143 seq(A360143(n),n=0..70) ;# _R. J. Mathar_, Mar 12 2023
%t A360143 Table[Sum[Binomial[2n+2k,n-k],{k,0,n}],{n,0,30}] (* _Harvey P. Dale_, Jul 23 2025 *)
%o A360143 (PARI) a(n) = sum(k=0, n, binomial(2*n+2*k, n-k));
%o A360143 (PARI) my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x*(2/(1+sqrt(1-4*x)))^4)))
%Y A360143 Cf. A001700, A032443, A108080, A360144.
%Y A360143 Cf. A000108, A000984, A002057.
%K A360143 nonn
%O A360143 0,2
%A A360143 _Seiichi Manyama_, Jan 27 2023