A261266 - OEIS

%I #19 Jan 30 2020 21:29:17

%S 1,2,8,44,264,1632,10256,65200,418144,2700224,17534208,114380928,

%T 748988928,4920379648,32413343488,214038123264,1416349369856,

%U 9389756730368,62352450867200,414660440811520,2761261291024384

%N Expansion of ((x-1/2)*(1/sqrt(8*x^2-8*x+1)+1)-x)/(x-1).

%H G. C. Greubel, <a href="/A261266/b261266.txt">Table of n, a(n) for n = 0..1000</a>

%H D. Kruchinin and V. Kruchinin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Kruchinin/kruch9.html">A Generating Function for the Diagonal T2n,n in Triangles</a>, Journal of Integer Sequence, Vol. 18 (2015), article 15.4.6.

%F a(n) = Sum_{l=0..n}(C(n,l)*Sum_{i=l..n}(C(i+l-1,i)*C(n-l,n-i))).

%F a(n) ~ 2^((3*n-1)/2) * (1+sqrt(2))^(n-1/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Aug 13 2015

%F D-finite with recurrence: n*a(n) +(-11*n+8)*a(n-1) +2*(17*n-28)*a(n-2) +8*(-5*n+12)*a(n-3) +16*(n-3)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2020

%p S := (n,k) -> simplify(binomial(2*k-1,k)*hypergeom([2*k,k-n],[k+1],-1)):

%p a := (n) -> add(binomial(n,k)*S(n,k), k=0..n):

%p seq(a(n), n=0..20); # _Peter Luschny_, Aug 13 2015

%t CoefficientList[Series[((x - 1/2)*(1/Sqrt[8*x^2 - 8*x + 1] + 1) - x)/(x - 1), {x, 0, 50}], x] (* _G. C. Greubel_, Jun 04 2017 *)

%o (Maxima)

%o a(n):=sum(binomial(n,l)*sum(binomial(i+l-1,i)*binomial(n-l,n-i),i,l,n),l,0,n);

%o (PARI) x='x+O('x^50); Vec(((x-1/2)*(1/sqrt(8*x^2-8*x+1)+1)-x)/(x-1)) \\ _G. C. Greubel_, Jun 04 2017

%Y Cf. A118376.

%K nonn

%O 0,2

%A _Vladimir Kruchinin_, Aug 13 2015