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A188441 Partial sums of binomial(2n,n)*binomial(3n,n) (A006480).

%I A188441 #28 Feb 28 2021 06:33:15
%S A188441 1,7,97,1777,36427,793183,17946319,417019279,9882531049,237755962549,
%T A188441 5788752753889,142315748216929,3527047510738129,88005145583604529,
%U A188441 2208577811494332529,55703557596868964209,1411049022002884046539
%N A188441 Partial sums of binomial(2n,n)*binomial(3n,n) (A006480).
%H A188441 Seiichi Manyama, <a href="/A188441/b188441.txt">Table of n, a(n) for n = 0..701</a>
%F A188441 a(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(3*k,k).
%F A188441 Recurrence: (n+2)^2*a(n+2)-(28*n^2+85*n+64)*a(n+1)+3*(9*n^2+27*n+20)*a(n) = 0.
%F A188441 G.f.: F(1/3,2/3;1;27*x)/(1-x), where F(a1,a2;b1;z) is a hypergeometric series.
%F A188441 a(n) ~ 3^(3*n+7/2) / (52*Pi*n). - _Vaclav Kotesovec_, Mar 02 2014
%F A188441 a(n) = hypergeom([1/3, 2/3], [1], 27) - hypergeom([1, n+4/3, n+5/3], [n+2, n+2], 27)*multinomial(n+1, n+1, n+1). - _Vladimir Reshetnikov_, Oct 12 2016
%t A188441 Table[Sum[Binomial[2k, k]Binomial[3k,k],{k,0,n}],{n,0,16}]
%t A188441 Round@Table[Hypergeometric2F1[1/3, 2/3, 1, 27] - HypergeometricPFQ[{1, n + 4/3, n + 5/3}, {n + 2, n + 2}, 27] Multinomial[n + 1, n + 1, n + 1], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 12 2016 *)
%t A188441 Accumulate[Table[Binomial[2n,n]Binomial[3n,n],{n,0,20}]] (* _Harvey P. Dale_, Oct 27 2020 *)
%o A188441 (Maxima) makelist(sum(binomial(2*k,k)*binomial(3*k,k),k,0,n),n,0,16);
%o A188441 (PARI) a(n) = sum(k=0, n, binomial(2*k,k)*binomial(3*k,k)); \\ _Michel Marcus_, Oct 13 2016
%Y A188441 Cf. A006480.
%K A188441 nonn,easy
%O A188441 0,2
%A A188441 _Emanuele Munarini_, Apr 14 2011