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A349768 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * binomial(2*k,k) / (k+1).

%I A349768 #18 Mar 02 2023 10:56:58
%S A349768 1,3,19,173,1881,22655,291775,3940725,55149025,793387235,11668476579,
%T A349768 174735112997,2656296912361,40897718776647,636588467802679,
%U A349768 10002872642155085,158483629611962025,2529389028336106475,40631849127696017275,656509442594976984405,10663184061320964941761
%N A349768 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * binomial(2*k,k) / (k+1).
%F A349768 From _Vaclav Kotesovec_, Nov 29 2021: (Start)
%F A349768 D-finite recurrence: n*(n+1)*(2*n - 3)*a(n) = (2*n - 1)*(19*n^2 - 37*n + 12)*a(n-1) - (2*n - 3)*(19*n^2 - 39*n + 14)*a(n-2) + (n-3)*(n-2)*(2*n - 1)*a(n-3).
%F A349768 a(n) ~ sqrt(5) * phi^(6*n + 3) / (8*Pi*n^2), where phi = A001622 is the golden ratio. (End)
%F A349768 D-finite with recurrence n*(n+1)*a(n) +(n+1)*(n-4)*a(n-1) +2*(-171*n^2 +512*n -388)*a(n-2) +2*(9*n^2 +296*n -796)*a(n-3) +(341*n^2 -2425*n +4320)*a(n-4) -19*(n-4)*(n-5)*a(n-5)=0. - _R. J. Mathar_, Mar 02 2023
%p A349768 A349768 := proc(n)
%p A349768     hypergeom([1/2,-n,n+1],[1,2],-4) ;
%p A349768     simplify(%) ;
%p A349768 end proc:
%p A349768 seq(A349768(n),n=0..20) ; # _R. J. Mathar_, Mar 02 2023
%t A349768 Table[Sum[Binomial[n, k] Binomial[n + k, k] Binomial[2 k, k]/(k + 1), {k, 0, n}], {n, 0, 20}]
%t A349768 Table[HypergeometricPFQ[{1/2, -n, n + 1}, {1, 2}, -4], {n, 0, 20}]
%o A349768 (PARI) a(n) = sum(k=0, n, binomial(n,k)*binomial(n+k,k)*binomial(2*k,k)/(k+1)); \\ _Michel Marcus_, Nov 29 2021
%Y A349768 Cf. A000108, A086618, A086621, A243945.
%K A349768 nonn,easy
%O A349768 0,2
%A A349768 _Ilya Gutkovskiy_, Nov 29 2021