OFFSET
0,2
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)
REFERENCES
C. Dement, Floretion Integer Sequences (work in progress)
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..400
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
EXAMPLE
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)
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 *)
PROG
(PARI) x='x + O('x^50); Vec(serlaplace((1 - log(sqrt(1 - 2*x)))/sqrt(1 - 2*x))) \\ G. C. Greubel, Feb 08 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Creighton Dement, Nov 14 2005
EXTENSIONS
E.g.f. given by Vladeta Jovovic
More terms from Robert G. Wilson v, Nov 15 2005
STATUS
approved