A110491 Expansion of e.g.f.: sqrt(1+2x)/sqrt(1-2x).

1, 2, 4, 24, 144, 1440, 14400, 201600, 2822400, 50803200, 914457600, 20118067200, 442597478400, 11507534438400, 299195895398400, 8975876861952000, 269276305858560000, 9155394399191040000, 311283409572495360000

Offset: 0

Author

Paul Barry, Jul 22 2005

Keywords

• easy
• nonn

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

Crossrefs

Cf. A000246, A002867, A063886.

Programs

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 *)

PARI

my(x='x+O('x^25)); Vec(serlaplace(sqrt(1+2*x)/sqrt(1-2*x))) \\ Michel Marcus, Aug 05 2021

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)

Sum_{n>=0} 1/a(n) = 1 + (StruveL(-1,1/2) + StruveL(0,1/2))*Pi/4, where StruveL is the modified Struve function. - Amiram Eldar, Aug 15 2025