A240599 - OEIS

%I #20 Dec 07 2021 02:32:53

%S 2,10,78,694,6570,64450,646870,6596334,68050578,708277498,7423267742,

%T 78238452390,828411940538,8805333110194,93899794491046,

%U 1004158446353374,10764586985847330,115642794779838442

%N Expansion of A(x) = x*B'(x)*(B(x)-x)/B(x)^2 where B(x)/x is g.f. of A027307.

%F a(n) = sum(k=1..n, 2^k*binomial(n-1,n-k)*sum(i=0..k, i*(-1)^(i-k)*binomial(k,i)*sum(j=0..n, ((-1)^j*2^(n-j)*(2*n-j+i-1)!)/(j!*(n-j)!*(n-j+i)!)))).

%F A(x) = x*B'(x)*(B(x)-x)/B(x)^2 where B(x) = x*((2*sqrt((x+3)/x)* sin(asin((sqrt(x)*(x+18))/(x+3)^(3/2))/3))/3-1/3).

%F Recurrence: n*(2*n-1)*(10*n^3 - 47*n^2 + 57*n - 11)*a(n) = 2*(110*n^5 - 627*n^4 + 1045*n^3 - 234*n^2 - 666*n + 345)*a(n-1) + (n-2)*(2*n-5)*(10*n^3 - 17*n^2 - 7*n + 9)*a(n-2). - _Vaclav Kotesovec_, Apr 16 2014

%F a(n) ~ sqrt(70*sqrt(5)-150) * ((11+5*sqrt(5))/2)^n / (20*sqrt(Pi*n)). - _Vaclav Kotesovec_, Apr 16 2014

%F Equivalently, a(n) ~ phi^(5*n - 2) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 07 2021

%t Table[Sum[2^k*Binomial[n-1, n-k]*Sum[i*(-1)^(i-k)*Binomial[k, i]*Sum[((-1)^j*2^(n-j)*(2*n-j+i-1)!)/(j!*(n-j)!*(n-j+i)!), {j, 0, n}], {i, 0, k}], {k, 1, n}],{n,1,20}] (* _Vaclav Kotesovec_, Apr 16 2014 after _Vladimir Kruchinin_ *)

%o (Maxima)

%o a(n):=sum(2^k*binomial(n-1,n-k)*sum(i*(-1)^(i-k)*binomial(k,i)*sum(((-1)^j*2^(n-j)*(2*n-j+i-1)!)/(j!*(n-j)!*(n-j+i)!),j,0,n),i,0,k),k,1,n)

%Y Cf. A027307.

%K nonn

%O 1,1

%A _Vladimir Kruchinin_, Apr 08 2014