A240688 - OEIS

%I #38 Feb 14 2022 11:23:04

%S 1,1,5,19,81,351,1553,6959,31489,143551,658305,3033471,14034177,

%T 65147135,303285505,1415422719,6620053505,31021657087,145613977601,

%U 684537354239,3222408929281,15187861143551,71663163121665

%N Expansion of -(x*sqrt(-4*x^2-4*x+1)-2*x^2-3*x) / ((x+1)*sqrt(-4*x^2-4*x+1)+ 4*x^3+8*x^2+3*x-1).

%H G. C. Greubel, <a href="/A240688/b240688.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)

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

%F A(x) = x*D'(x)/D(x) where D(x)=(1-sqrt(1-4*x-4*x^2))/(2*(1+x)) is g.f. of A052709.

%F a(n) ~ 2^(n-1/4) * (1+sqrt(2))^(n-1/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Apr 12 2014

%F a(n) = Sum_{i=0..n/2} binomial(n,i)*binomial(2*n-2*i-1,n-2*i). - _Vladimir Kruchinin_, Mar 10 2015

%F Conjecture: n*(n-1)*a(n) -(3*n-2)*(n-1)*a(n-1) +2*(-4*n^2+7*n-1)*a(n-2) -4*n*(n-2)*a(n-3)=0. - _R. J. Mathar_, Jun 14 2016

%F From _Peter Bala_, Feb 13 2022: (Start)

%F The o.g.f. A(x) satisfies the differential equation (8*x^4 + 20*x^3 + 14*x^2 + x - 1)*A(x)' + (8*x^3 + 12*x^2 + 6*x + 3)*A(x) - 2 = 0 with A(0) = 1.

%F n*a(n) = (n+2)*a(n-1) + (14*n-22)*a(n-2) + (20*n-48)*a(n-3) + (8*n-24)*a(n-4).

%F Mathar's conjectural third-order recurrence above can be verified using Maple's gfun:-rectodiffeq command.

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

%t CoefficientList[Series[-(x Sqrt[-4 x^2 - 4 x + 1] - 2 x^2 - 3 x) / ((x + 1) Sqrt[-4 x^2 - 4 x + 1] + 4 x^3 + 8 x^2 + 3 x - 1), {x, 0, 25}], x] (* _Vaclav Kotesovec_, Apr 12 2014 *)

%o (Maxima)

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

%o (PARI) x='x+O('x^50); Vec(-(x*sqrt(-4*x^2-4*x+1)-2*x^2-3*x) / ((x+1)*sqrt(-4*x^2-4*x+1)+ 4*x^3+8*x^2+3*x-1)) \\ _G. C. Greubel_, Apr 05 2017

%Y Cf. A052709.

%K nonn,easy

%O 0,3

%A _Vladimir Kruchinin_, Apr 10 2014