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A293490 a(n) = Sum_{k=0..n} binomial(2*k, k)*binomial(2*n-k, n).

%I A293490 #22 Feb 08 2021 07:10:24
%S A293490 1,4,18,84,400,1932,9436,46512,231066,1155660,5813808,29396952,
%T A293490 149305884,761282032,3894953640,19987999696,102847396416,530446714812,
%U A293490 2741576339716,14196136939600,73631851898220,382483602131400,1989514312826400,10361255764532400,54020655931542300,281933439875693424
%N A293490 a(n) = Sum_{k=0..n} binomial(2*k, k)*binomial(2*n-k, n).
%C A293490 Main diagonal of iterated partial sums array of central binomial coefficients (starting with the first partial sums).
%F A293490 a(n) = [x^n] 1/(sqrt(1 - 4*x)*(1 - x)^(n+1)).
%F A293490 a(n) = [x^n] 1/((1 - x)^(n+1)*(1 - 2*x/(1 - x/(1 - x/(1 - x/(1 - ...)))))), a continued fraction.
%F A293490 a(n) = 4^n*Gamma(n+1/2)*2F1(-n,n+1; 1/2-n; 1/4)/(sqrt(Pi)*Gamma(n+1)).
%F A293490 From _Vaclav Kotesovec_, Oct 16 2017: (Start)
%F A293490 D-finite with recurrence: 3*(n-1)*n*a(n) = 14*(n-1)*(2*n-1)*a(n-1) - 4*(4*n-5)*(4*n-3)*a(n-2).
%F A293490 a(n) ~ 2^(4*n + 3/2) / (3^(n + 1/2) * sqrt(Pi*n)).
%F A293490 (End)
%F A293490 G.f.: 1/(sqrt(2*sqrt(1-4*x)-1)*sqrt(1-4*x)). - _Alexander M. Haupt_, Jun 24 2018
%t A293490 Table[Sum[Binomial[2 k, k] Binomial[2 n - k, n], {k, 0, n}], {n, 0, 25}]
%t A293490 Table[SeriesCoefficient[1/(Sqrt[1 - 4 x] (1 - x)^(n + 1)), {x, 0, n}], {n, 0, 25}]
%t A293490 Table[SeriesCoefficient[(1/(1 - x)^(n + 1)) 1/(1 - 2 x/(1 + ContinuedFractionK[-x, 1, {k, 1, n}])), {x, 0, n}], {n, 0, 25}]
%t A293490 CoefficientList[Series[1/(Sqrt[2 Sqrt[1-4 x]-1] Sqrt[1-4 x]),{x,0,25}],x] (* _Alexander M. Haupt_, Jun 24 2018 *)
%o A293490 (GAP) A293490 := Concatenation([1], List([1..3*10^2],n -> Sum([0..n],k -> Binomial(2*k, k)*(Binomial(2*n - k, n))))); # _Muniru A Asiru_, Oct 15 2017
%o A293490 (PARI) a(n) = sum(k=0, n, binomial(2*k, k)*binomial(2*n-k, n)); \\ _Michel Marcus_, Oct 15 2017
%Y A293490 Cf. A000984, A002426, A006134, A026375, A270447.
%K A293490 nonn
%O A293490 0,2
%A A293490 _Ilya Gutkovskiy_, Oct 10 2017