A242586 - OEIS

%I #39 Aug 05 2022 06:23:11

%S 1,2,6,23,98,437,1995,9242,43258,204053,968441,4619012,22120631,

%T 106300508,512321438,2475395303,11986728458,58156146653,282640193313,

%U 1375737276788,6705522150973,32724071280518,159878425878848

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

%C Binomial transform of A088218.

%H Harvey P. Dale, <a href="/A242586/b242586.txt">Table of n, a(n) for n = 0..1000</a>

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

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

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

%F a(n) = hypergeom([1/2,-n],[1],-4)/2 + 1/2. - _Peter Luschny_, May 18 2014

%F 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

%F 2*a(n) = 1 + A026375(n). - _R. J. Mathar_, Jan 26 2020

%F From _Peter Bala_, Jan 09 2022: (Start)

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

%F 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.

%F 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)

%p a := n -> hypergeom([1/2,-n],[1], -4)/2 + 1/2;

%p seq(round(evalf(a(n),32)), n=0..22); # _Peter Luschny_, May 18 2014

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

%o (Maxima)

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

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

%K nonn,easy

%O 0,2

%A _Vladimir Kruchinin_, May 18 2014