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A346664 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(4*k,k) / (3*k + 1).

%I A346664 #16 Aug 17 2023 05:05:11
%S A346664 1,0,3,12,73,453,2985,20373,142933,1024302,7466211,55182240,412586977,
%T A346664 3115105321,23717115513,181884676827,1403719428485,10894049061956,
%U A346664 84967420574247,665643698649684,5235570329071893,41328838600501830,327315349579739619,2600034901186102182
%N A346664 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(4*k,k) / (3*k + 1).
%C A346664 Inverse binomial transform of A002293.
%H A346664 Seiichi Manyama, <a href="/A346664/b346664.txt">Table of n, a(n) for n = 0..1000</a>
%F A346664 G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^2 * A(x)^4.
%F A346664 G.f.: Sum_{k>=0} ( binomial(4*k,k) / (3*k + 1) ) * x^k / (1 + x)^(k+1).
%F A346664 a(n) ~ 229^(n + 3/2) / (2048 * sqrt(2*Pi) * n^(3/2) * 3^(3*n + 3/2)). - _Vaclav Kotesovec_, Jul 30 2021
%F A346664 D-finite with recurrence +3*n*(3*n-1)*(3*n+1)*a(n) -74*n*(2*n-1) *(n-1)*a(n-1) -6*(n-1) *(101*n^2 -202*n +105)*a(n-2) -330*(n-1) *(n-2)*(2*n-3) *a(n-3) -229*(n-1)*(n-2) *(n-3)*a(n-4)=0. - _R. J. Mathar_, Aug 17 2023
%p A346664 A346664 := proc(n)
%p A346664     add( (-1)^(n-k)*binomial(n,k)*binomial(4*k,k)/(3*k+1),k=0..n) ;
%p A346664 end proc:
%p A346664 seq(A346664(n),n=0..80); # _R. J. Mathar_, Aug 17 2023
%t A346664 Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 23}]
%t A346664 nmax = 23; 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 A346664 nmax = 23; CoefficientList[Series[Sum[(Binomial[4 k, k]/(3 k + 1)) x^k/(1 + x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
%t A346664 Table[(-1)^n HypergeometricPFQ[{1/4, 1/2, 3/4, -n}, {2/3, 1, 4/3}, 256/27], {n, 0, 23}]
%o A346664 (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n,k)*binomial(4*k,k)/(3*k+1)); \\ _Michel Marcus_, Jul 28 2021
%Y A346664 Cf. A002293, A005043, A188678, A346628, A346646, A346665, A346666, A346667, A346668.
%K A346664 nonn
%O A346664 0,3
%A A346664 _Ilya Gutkovskiy_, Jul 27 2021