%I #17 Aug 17 2023 05:02:11

%S 1,2,7,38,257,1935,15505,129519,1115061,9823160,88121887,802227794,

%T 7392428009,68819554003,646276497617,6114880542117,58237420303109,

%U 557850829527246,5370956411708779,51947475492561014,504492516832543885,4917564488572565160

%N a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k) / (3*k + 1).

%C Binomial transform of A002293.

%H Seiichi Manyama, <a href="/A346646/b346646.txt">Table of n, a(n) for n = 0..500</a>

%F G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^2 * A(x)^4.

%F G.f.: Sum_{k>=0} ( binomial(4*k,k) / (3*k + 1) ) * x^k / (1 - x)^(k+1).

%F a(n) ~ 283^(n + 3/2) / (2048 * sqrt(2*Pi) * n^(3/2) * 3^(3*n + 3/2)). - _Vaclav Kotesovec_, Jul 30 2021

%F D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n) -2*(2*n-1) *(91*n^2 -91*n +24)*a(n-1) +6*(n-1) *(155*n^2 -310*n +167)*a(n-2) -438*(n-1) *(n-2)*(2*n-3) *a(n-3) +283*(n-1)*(n-2) *(n-3)*a(n-4)=0. - _R. J. Mathar_, Aug 17 2023

%p A346646 := proc(n)

%p hypergeom([-n,1/4,1/2,3/4],[2/3,1,4/3],-256/27) ;

%p simplify(%) ;

%p end proc:

%p seq(A346646(n),n=0..40) ; # _R. J. Mathar_, Jan 10 2023

%t Table[Sum[Binomial[n, k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 21}]

%t nmax = 21; A[_] = 0; Do[A[x_] = 1/(1 - x) + x (1 - x)^2 A[x]^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%t nmax = 21; CoefficientList[Series[Sum[(Binomial[4 k, k]/(3 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]

%t Table[HypergeometricPFQ[{1/4, 1/2, 3/4, -n}, {2/3, 1, 4/3}, -256/27], {n, 0, 21}]

%o (PARI) a(n) = sum(k=0, n, binomial(n,k)*binomial(4*k,k)/(3*k + 1)); \\ _Michel Marcus_, Jul 26 2021

%Y Cf. A002293, A007317, A104859, A188687, A226974, A346647, A346648, A346649, A346650.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Jul 26 2021