A242586 Expansion of 1/(2*sqrt(1-x))*(1/sqrt(1-x)+1/(sqrt(1-5*x))).

1, 2, 6, 23, 98, 437, 1995, 9242, 43258, 204053, 968441, 4619012, 22120631, 106300508, 512321438, 2475395303, 11986728458, 58156146653, 282640193313, 1375737276788, 6705522150973, 32724071280518, 159878425878848

Offset

0,2

Comments

Binomial transform of A088218.

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Formula

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

G.f. A(x) = x*F'(x)/F(x), where F(x) is g.f. of A007317.

a(n) = T(2*n,n) for n>0, where T(n,k) is triangle of A105477.

a(n) = hypergeom([1/2,-n],[1],-4)/2 + 1/2. - Peter Luschny, May 18 2014

D-finite with recurrence: n*a(n) + (-7*n+4)*a(n-1) + (11*n-14)*a(n-2) + 5*(-n+2)*a(n-3) = 0. - R. J. Mathar, May 23 2014

2*a(n) = 1 + A026375(n). - R. J. Mathar, Jan 26 2020

From Peter Bala, Jan 09 2022: (Start)

a(n) = [x^n] ( x + 1/(1 - x) )^n.

a(0) = 1, a(1) = 2 and n*a(n) = 3*(2*n-1)*a(n-1) - 5*(n-1)*a(n-2) - 1 for n >= 2.

The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. (End)

Maple

a := n -> hypergeom([1/2, -n], [1], -4)/2 + 1/2;
seq(round(evalf(a(n), 32)), n=0..22); # Peter Luschny, May 18 2014

Mathematica

CoefficientList[Series[1/(2Sqrt[1-x]) (1/Sqrt[1-x]+1/Sqrt[1-5x]), {x, 0, 30}], x] (* Harvey P. Dale, Mar 19 2020 *)

Prog

(Maxima)
a(n):=sum(binomial(2*j-1, j)*binomial(n, j), j, 0, n);

Crossrefs

Cf. A007317, A088218, A105477, A110166, A246437.

Sequence in context: A280768 A370183 A278301 * A196018 A009449 A233106

Adjacent sequences: A242583 A242584 A242585 * A242587 A242588 A242589

Keyword

nonn,easy

Author

Vladimir Kruchinin, May 18 2014

Status

approved