OFFSET
0,2
COMMENTS
Row sums of exponential Riordan array [1, arctanh(2x)]. - Paul Barry, Apr 17 2008
Conjecture: {a(n-1), n>=1} is the T-transform of A093178, where T maps a sequence {b(n), n>=1} to the sequence {c(n)} defined by c(n) = det(M_n), where M_n is the n X n matrix with elements M_n(i,j) = b(2*j) for i>j and M_n(i,j) = b(i+j-1) for i<=j. - Lechoslaw Ratajczak, Aug 04 2021
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
E.g.f.: sqrt((1+2x)/(1-2x)); a(n)=2*n!*binomial(n-1, floor((n-1)/2))+0^n.
The sequence 0,1,0,2,0,4,... has e.g.f. arctanh(x). - Paul Barry, Apr 17 2008
D-finite with recurrence a(n) -2*a(n-1) -4*(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Sep 20 2012
a(n) ~ 2^(n+1)*n^n/exp(n). - Vaclav Kotesovec, Sep 25 2013
a(n) = 2^n*binomial(1/2,n)*n!*2F1(1/2,-n;3/2-n;-1). - Benedict W. J. Irwin, Jun 06 2016
From Robert Israel, Jun 08 2016: (Start)
a(n) = n! * A063886(n).
E.g.f. satisfies 2*g(x)+(4*x^2-1)*g'(x) = 0, from which Mathar's recurrence follows. (End)
MAPLE
S:= series(sqrt(1+2*x)/sqrt(1-2*x), x, 31):
seq(coeff(S, x, j)*j!, j=0..30); # Robert Israel, Jun 08 2016
MATHEMATICA
With[{nn=20}, CoefficientList[Series[Sqrt[1+2x]/Sqrt[1-2x], {x, 0, nn}], x] Range[0, nn]!] (* or *) Join[{1}, Table[2n!Binomial[n-1, Floor[(n-1)/2]], {n, 20}]] (* Harvey P. Dale, Nov 11 2011 *)
Table[2^n Binomial[1/2, n] n! Hypergeometric2F1[1/2, -n, 3/2 - n, -1], {n, 0, 20}] (* Benedict W. J. Irwin, Jun 06 2016 *)
PROG
(PARI) my(x='x+O('x^25)); Vec(serlaplace(sqrt(1+2*x)/sqrt(1-2*x))) \\ Michel Marcus, Aug 05 2021
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jul 22 2005
STATUS
approved