A277871 - OEIS

%I #25 Apr 10 2017 23:00:24

%S 1,4,16,66,279,1203,5275,23474,105853,483108,2229253,10390691,

%T 48879588,231879456,1108473015,5335987930,25849521109,125945214309,

%U 616833862018,3035286848660,14999774773110,74413424196360,370463714276625,1850251796668899

%N a(n) = Sum_{i=0..n+1} binomial(2*n-i,n-i+1)*Catalan(i).

%C T(2*n+1,n) is diagonal of triangle A125177.

%H Seiichi Manyama, <a href="/A277871/b277871.txt">Table of n, a(n) for n = 0..1381</a>

%F G.f.: (4*x*(1-sqrt(1-2*(1-sqrt(1-4*x)))))/(1-sqrt(1-4*x))^3/sqrt(1-4*x)-1/x.

%F a(n) ~ 2^(4*n+1/2) / (sqrt(Pi) * n^(3/2) * 3^(n-3/2)). - _Vaclav Kotesovec_, Nov 05 2016

%t Table[Sum[Binomial[2*i, i]*Binomial[2*n-i, n-i+1]/(i+1), {i, 0, n+1}], {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 05 2016 *)

%o (Maxima)

%o a(n):=sum((binomial(2*i,i)*binomial(2*n-i,n-i+1))/(i+1),i,0,n+1);

%o (PARI) x='x+O('x^50); Vec((4*x*(1-sqrt(1-2*(1-sqrt(1-4*x)))))/(1-sqrt(1-4*x))^3/sqrt(1-4*x)-1/x) \\ _G. C. Greubel_, Apr 09 2017

%Y Cf. A000108, A125177.

%K nonn

%O 0,2

%A _Vladimir Kruchinin_, Nov 02 2016