A187071 Expansion of d/dx arctan(x*A001003(x)).

1, 2, 8, 40, 206, 1084, 5802, 31440, 171946, 947132, 5247010, 29203928, 163176586, 914744612, 5142354178, 28978786976, 163652047834, 925925993132, 5247514156418, 29783577676840, 169270380108906, 963186164033652

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..199 from Nathaniel Johnston)

Formula

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

G.f.: d/dx arctan(x*2/(1+x+sqrt(1-6*x+x^2))) = (sqrt(x^2-6*x+1)-x+3) / (4*sqrt(x^2-6*x+1) * ((-sqrt(x^2-6*x+1)+x+1)^2/16+1)).

Recurrence: 5*n*(17*n-24)*a(n) = (544*n^2 - 1023*n + 385)*a(n-1) - (323*n^2 - 643*n + 224)*a(n-2) + 2*(119*n^2 - 236*n + 91)*a(n-3) - 2*(n-2)*(17*n-7)*a(n-4). - Vaclav Kotesovec, Oct 24 2012

a(n) ~ sqrt(252+179*sqrt(2))*(3+2*sqrt(2))^n/(34*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 24 2012

Mathematica

CoefficientList[Series[(Sqrt[x^2-6*x+1]-x+3)/(4*Sqrt[x^2-6*x+1]*((-Sqrt[x^2-6*x+1]+x+1)^2/16+1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

Prog

(Maxima)
a(n):=sum((sum((-1)^j*2^(-j)*binomial(n+1, j)*binomial(2*n-k-j, n), j, 0, n-k))*(2^(n-k-1))*(1-(-1)^(k+1))*(-1)^(k/2), k, 0, n);
(PARI) x='x+O('x^50); Vec((sqrt(x^2-6*x+1)-x+3) / (4*sqrt(x^2-6*x+1)*((-sqrt(x^2-6*x+1)+x+1)^2/16+1))) \\ G. C. Greubel, Mar 26 2017

Crossrefs

Cf. A001003.

Sequence in context: A186947 A071007 A027617 * A154626 A003305 A076625

Adjacent sequences: A187068 A187069 A187070 * A187072 A187073 A187074

Keyword

nonn

Author

Vladimir Kruchinin, Apr 10 2011

Status

approved