login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

a(n) = Sum_{k=0..n}((binomial(2*k,k)/(k+1)*binomial(2*n+2,n-k))).
1

%I #18 Apr 10 2017 02:52:23

%S 1,5,23,105,484,2267,10821,52705,262010,1328768,6867266,36115455,

%T 192954358,1045481465,5735154907,31802349105,178010615678,

%U 1004542994462,5709066033900,32646940202200,187701954810320

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

%C Binomial transform of Catalan numbers.

%H G. C. Greubel, <a href="/A270530/b270530.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1/(2*x*sqrt(1-4*x))+(-sqrt(((5*x+2*sqrt(1-4*x)-2))/(x^3*(4-16*x)))).

%F a(n) ~ 5^(2*n + 7/2) / (3^(3/2) * sqrt(Pi) * n^(3/2) * 2^(2*n+4)). - _Vaclav Kotesovec_, Mar 18 2016

%F Conjecture: 2*n*(2*n+3)*(n+1)*a(n) -n*(77*n^2+27*n-4)*a(n-1) +(549*n^3-987*n^2+686*n-168)*a(n-2) -20*(2*n-3)*(43*n^2-104*n+70)*a(n-3) +500*(2*n-5)*(n-2)*(2*n-3)*a(n-4)=0. - _R. J. Mathar_, Jun 07 2016

%F Conjecture: 2*n*(2*n+3)*(n+3)*(n+1)*a(n) -n*(57*n^3+228*n^2+107*n+8)*a(n-1) +4*(2*n-1) *(33*n^3+99*n^2-88*n+36)*a(n-2) -100*(n-1)*(2*n-1)*(2*n-3)*(n+4)*a(n-3)=0. - _R. J. Mathar_, Jun 07 2016

%p A270530 := proc(n)

%p add(binomial(2*k,k)/(k+1)*binomial(2*n+2,n-k),k=0..n) ;

%p end proc: # _R. J. Mathar_, Jun 07 2016

%t CoefficientList[Series[1/(2*x*Sqrt[1 - 4*x]) + (-Sqrt[((5*x + 2*Sqrt[1 - 4*x] - 2))/(x^3*(4 - 16*x))]), {x,0,50}], x] (* _G. C. Greubel_, Apr 09 2017 *)

%o (Maxima)

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

%o makelist(coeff(taylor(1/(2*x*sqrt(1-4*x))+(-sqrt(((5*x+2*sqrt(1-4*x)-2))/(x^3*(4-16*x)))),x,0,10),x,n),n,0,10);

%o (PARI) x='x+O('x^50); Vec(1/(2*x*sqrt(1-4*x))+(-sqrt(((5*x+2*sqrt(1-4*x)-2))/(x^3*(4-16*x))))) \\ _G. C. Greubel_, Apr 09 2017

%Y Cf. A000108, A007317, A270447.

%K nonn

%O 0,2

%A _Vladimir Kruchinin_, Mar 18 2016