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A188946 Binomial partial sums of binomial(2n,n)*binomial(3n,n) (A006480).

%I A188946 #18 Sep 08 2022 08:45:56
%S A188946 1,7,103,1969,41935,947737,22248409,536310271,13183283743,
%T A188946 328970388985,8307368234473,211822788505951,5444571611722369,
%U A188946 140892128574440887,3667015053678269095,95918056089104563489,2519845343307697266943
%N A188946 Binomial partial sums of binomial(2n,n)*binomial(3n,n) (A006480).
%F A188946 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)*binomial(3*k,k).
%F A188946 Recurrence: (n+3)^2*a(n+3)-(30*n^2+150*n+187)*a(n+2)+57*(n+2)^2*a(n+1)-28*(n+1)*(n+2)*a(n)=0.
%F A188946 E.g.f.: exp(x)*F(1/3,2/3;1,1;27*x), where F(a1,a2;b1;z) is a hypergeometric series.
%F A188946 a(n) = hypergeom([1/3, 2/3, -n], [1, 1], -27). - _Vladimir Reshetnikov_, Oct 15 2017
%F A188946 a(n) ~ 2^(2*n+1) * 7^(n+1) / (3^(5/2)*Pi*n). - _Vaclav Kotesovec_, Nov 27 2017
%t A188946 Table[Sum[Binomial[n,k]Binomial[2k,k]Binomial[3k,k],{k,0,n}],{n,0,16}]
%t A188946 Table[HypergeometricPFQ[{1/3, 2/3, -n}, {1, 1}, -27], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 15 2017 *)
%o A188946 (Maxima) makelist(sum(binomial(n,k)*binomial(2*k,k)*binomial(3*k,k),k,0,n),n,0,16);
%o A188946 (PARI) a(n) = sum(k=0, n, binomial(n,k)*binomial(2*k,k)*binomial(3*k,k)); \\ _Michel Marcus_, Oct 15 2017
%o A188946 (Magma) [&+[Binomial(n, k)*Binomial(2*k, k)*Binomial(3*k, k): k in [0..n]]: n in [0.. 18]]; // _Vincenzo Librandi_, Oct 16 2017
%Y A188946 Cf. A006480, A188441, A188918, A152297.
%K A188946 nonn,easy
%O A188946 0,2
%A A188946 _Emanuele Munarini_, Apr 14 2011