A190733 Expansion of (4*x+2)/(1+sqrt(1-4*x-4*x^2)).

1, 3, 5, 15, 49, 175, 657, 2559, 10241, 41855, 173953, 732927, 3123457, 13439743, 58307841, 254779391, 1120247809, 4952864767, 22005184513, 98196398079, 439923990529, 1977917169663, 8921667641345, 40361657696255, 183092192411649, 832634240106495, 3795237359190017

Offset: 0

Author

Dmitry Kruchinin, May 26 2011

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100

Programs

Mathematica

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

Maxima

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

PARI

x='x+O('x^66); /* that many terms */
Vec((4*x+2)/(1+sqrt(1-4*x-4*x^2))) /* show terms */ /* Joerg Arndt, May 27 2011 */

Formula

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

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