A190733 - OEIS

%I #26 Jan 30 2020 21:29:16

%S 1,3,5,15,49,175,657,2559,10241,41855,173953,732927,3123457,13439743,

%T 58307841,254779391,1120247809,4952864767,22005184513,98196398079,

%U 439923990529,1977917169663,8921667641345,40361657696255,183092192411649,832634240106495,3795237359190017

%N Expansion of (4*x+2)/(1+sqrt(1-4*x-4*x^2)).

%H Vincenzo Librandi, <a href="/A190733/b190733.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = (n+1)*sum(k=1..n+1,binomial(k,n-k+1)*Catalan(k-1)/k).

%F D-finite with recurrence: (n+1)*a(n) +(-n+1)*a(n-1) +14*(-n+2)*a(n-2) +20*(-n+3)*a(n-3) +8*(-n+4)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2020

%t CoefficientList[Series[(4x+2)/(1+Sqrt[1-4x-4x^2]),{x,0,40}],x] (* _Harvey P. Dale_, Mar 20 2015 *)

%o (Maxima)

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

%o (PARI) x='x+O('x^66); /* that many terms */

%o Vec((4*x+2)/(1+sqrt(1-4*x-4*x^2))) /* show terms */ /* _Joerg Arndt_, May 27 2011 */

%K nonn

%O 0,2

%A _Dmitry Kruchinin_, May 26 2011