A114160 E.g.f. is A(x) = (1-log(B(x)))/B(x), where B(x) = sqrt(1-2*x).

1, 2, 7, 38, 281, 2634, 29919, 399342, 6125265, 106156530, 2051433495, 43734832470, 1019650457385, 25807495577850, 704708234182575, 20649996837971550, 646340185330747425, 21521124899877175650, 759572031366463998375, 28325808256035867711750, 1112907316518036732317625

Offset: 0

Author

Creighton Dement, Nov 14 2005

Keywords

• nonn

Comments

From John M. Campbell, May 20 2011: (Start)

a(n) is the determinant of the n X n matrix of the form:

|2 1 1 1 ... 1 |

|1 4 1 1 ... 1 |

|1 1 6 1 ... 1 |

|1 1 1 8 ... 1 |

|... ... 1 |

|1 1 1 1 2n-2 1 |

|1 1 1 1 1 2n |

See examples. (End)

Examples

			From _John M. Campbell_, May 20 2011: (Start)
Det[{
{2,1,1,1,1,1},
{1,4,1,1,1,1},
{1,1,6,1,1,1},
{1,1,1,8,1,1},
{1,1,1,1,10,1},
{1,1,1,1,1,12}}] = 29919 = a(6), and
Det[{
{2,1,1,1,1,1,1},
{1,4,1,1,1,1,1},
{1,1,6,1,1,1,1},
{1,1,1,8,1,1,1},
{1,1,1,1,10,1,1},
{1,1,1,1,1,12,1},
{1,1,1,1,1,1,14}}] = 399342 = a(7).
(End)
		

References

• C. Dement, Floretion Integer Sequences (work in progress)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A114161.

Programs

Mathematica

Range[0, 18]! CoefficientList[ Series[(1 - Log[Sqrt[1 - 2x]])/Sqrt[(1 - 2x)], {x, 0, 18}], x] (* or *)
f[n_] := FullSimplify[ 2^(n-1)*Gamma[n + 1/2]/Sqrt[Pi]*(PolyGamma[n + 1/2] + EulerGamma + Log[4] + 2)]; Table[f[n], {n, 0, 18}] (* Robert G. Wilson v *)
twox[x_, y_] := If[x == y, 2*x, 1]; a[n_] := Det[Array[twox[#1, #2] &, {n, n}]]; Join[{1}, Table[a[n], {n, 1, 10}]] (* John M. Campbell, May 20 2011 *)

PARI

my(x='x + O('x^50)); Vec(serlaplace((1 - log(sqrt(1 - 2*x)))/sqrt(1 - 2*x))) \\ G. C. Greubel, Feb 08 2017

Formula

a(n) = A001147(n) + A004041(n-1) = 2^n*Gamma(n+1/2)/Pi^(1/2)*(1/2*Psi(n+1/2)+1/2*gamma+log(2)+1). - Vladeta Jovovic

Extensions

E.g.f. given by Vladeta Jovovic

More terms from Robert G. Wilson v, Nov 15 2005