This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A110491 #49 Aug 15 2025 13:03:33
%S A110491 1,2,4,24,144,1440,14400,201600,2822400,50803200,914457600,
%T A110491 20118067200,442597478400,11507534438400,299195895398400,
%U A110491 8975876861952000,269276305858560000,9155394399191040000,311283409572495360000
%N A110491 Expansion of e.g.f.: sqrt(1+2x)/sqrt(1-2x).
%C A110491 Row sums of exponential Riordan array [1, arctanh(2x)]. - _Paul Barry_, Apr 17 2008
%C A110491 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
%H A110491 Vincenzo Librandi, <a href="/A110491/b110491.txt">Table of n, a(n) for n = 0..200</a>
%F A110491 E.g.f.: sqrt((1+2x)/(1-2x)); a(n)=2*n!*binomial(n-1, floor((n-1)/2))+0^n.
%F A110491 The sequence 0,1,0,2,0,4,... has e.g.f. arctanh(x). - _Paul Barry_, Apr 17 2008
%F A110491 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
%F A110491 a(n) ~ 2^(n+1)*n^n/exp(n). - _Vaclav Kotesovec_, Sep 25 2013
%F A110491 a(n) = 2^n*binomial(1/2,n)*n!*2F1(1/2,-n;3/2-n;-1). - _Benedict W. J. Irwin_, Jun 06 2016
%F A110491 From _Robert Israel_, Jun 08 2016: (Start)
%F A110491 a(n) = n! * A063886(n).
%F A110491 E.g.f. satisfies 2*g(x)+(4*x^2-1)*g'(x) = 0, from which Mathar's recurrence follows. (End)
%F A110491 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
%p A110491 S:= series(sqrt(1+2*x)/sqrt(1-2*x),x,31):
%p A110491 seq(coeff(S,x,j)*j!,j=0..30); # _Robert Israel_, Jun 08 2016
%t A110491 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 *)
%t A110491 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 *)
%o A110491 (PARI) my(x='x+O('x^25)); Vec(serlaplace(sqrt(1+2*x)/sqrt(1-2*x))) \\ _Michel Marcus_, Aug 05 2021
%Y A110491 Cf. A000246, A002867, A063886.
%K A110491 easy,nonn
%O A110491 0,2
%A A110491 _Paul Barry_, Jul 22 2005