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A339710 a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*n + k, k)*2^k.

%I A339710 #34 May 11 2021 06:09:27
%S A339710 1,7,81,1051,14353,201807,2891409,41976627,615371169,9089130967,
%T A339710 135048608401,2016306678987,30224723308081,454603719479839,
%U A339710 6857319231939537,103694587800440931,1571449259865571137,23860205774602899111,362897293035114695121,5527773456878667951483
%N A339710 a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*n + k, k)*2^k.
%D A339710 Frits Beukers, Some Congruences for Apery Numbers, Mathematisch Instituut, University of Leiden, 1983, pages 1-2.
%H A339710 Seiichi Manyama, <a href="/A339710/b339710.txt">Table of n, a(n) for n = 0..838</a>
%H A339710 F. Beukers, <a href="https://doi.org/10.1016/0022-314X(85)90047-2">Some congruences for the Apery numbers</a>, Journal of Number Theory, Vol. 21, Issue 2, Oct. 1985, pp. 141-155. <a href="/A339710/a339710.pdf">local copy</a>
%F A339710 a(n) = 2F1([-n, 1 + 2*n], [1], -2), where 2F1 is the hypergeometric function. - _Stefano Spezia_, Dec 17 2020
%F A339710 From _Vaclav Kotesovec_, May 11 2021: (Start)
%F A339710 Recurrence: 3*n*(2*n - 1)*(26*n - 35)*a(n) = (2444*n^3 - 5734*n^2 + 3830*n - 729)*a(n-1) - (n-1)*(2*n - 3)*(26*n - 9)*a(n-2).
%F A339710 a(n) ~ sqrt(3/8 + 11/(8*sqrt(13))) * ((47 + 13*sqrt(13))/6)^n / sqrt(Pi*n). (End)
%t A339710 Table[Sum[Binomial[n,k]*Binomial[2n+k,k]*2^k,{k,0,n}],{n,0,20}] (* or *)
%t A339710 Table[Hypergeometric2F1[-n,1+2 n,1,-2],{n,0,20}] (* _Stefano Spezia_, Dec 17 2020 *)
%o A339710 (PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(2*n + k, k)*2^k); \\ _Michel Marcus_, Feb 12 2021
%Y A339710 Cf. A000079 (Sum(binomial(n, k))), A000984 (Sum(binomial(n, k)^2)), A026375 (Sum(binomial(n, k)*binomial(2*k, k))), A001850 (Sum(binomial(n, k)*binomial(n+k, k))), A005809 (Sum(binomial(n, k)*binomial(2*n, k))).
%Y A339710 Cf. A026000, A114496.
%K A339710 nonn
%O A339710 0,2
%A A339710 _Yifan Zhang_, Dec 13 2020
%E A339710 More terms from _Stefano Spezia_, Dec 17 2020