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%I M3126 #92 Sep 08 2022 08:44:35
%S 1,3,30,630,22680,1247400,97297200,10216206000,1389404016000,
%T 237588086736000,49893498214560000,12623055048283680000,
%U 3786916514485104000000,1329207696584271504000000,539658324813214230624000000,250941121038144617240160000000,132496911908140357902804480000000
%N a(n) = (2n+1)! / 2^n.
%C Denominators of coefficients of the Taylor series of sinh(sqrt(2*x))/(sqrt(2*x)). - J. Zurita (jrzurita(AT)inaoep.mx), Dec 01 2007
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A007019/b007019.txt">Table of n, a(n) for n = 0..90</a>
%H L. Carlitz, <a href="http://projecteuclid.org/euclid.pjm/1103035736">The inverse of the error function</a>, Pacific J. Math., 13 (1963), 459-470.
%H R. Flórez and L. Junes, <a href="http://www.emis.de/journals/INTEGERS/papers/l50/l50.Abstract.html">A relation between triangular numbers and prime numbers</a>, Integers 12 (2012), no. 1, 83-96.
%H Z. Usiskin, <a href="/A007018/a007018.pdf">Letter to N. J. A. Sloane, Oct. 1991</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InverseErf.html">Inverse Erf</a>
%F sin(x)*cosh(x) = Sum_{n>=0} (-1)^floor(n/2)*x^(2n+1)/a(n). - _Benoit Cloitre_, Feb 02 2002
%F a(n) = Product_{k=0..n-1} (A000217(n+1) - A000217(k)). - _Anton Zakharov_, Sep 14 2016
%F a(n) ~ sqrt(Pi)*2^(n+2)*n^(2*n+3/2)/exp(2*n). - _Ilya Gutkovskiy_, Sep 14 2016
%F a(n) = Product_{j=1..n} T(2j) (where T(k) is the k-th triangular number). For example: a(3) = T(2)*T(4)*T(6) (that is, 630 = 3*10*21). - _Rigoberto Florez_, Aug 26 2018
%F From _Amiram Eldar_, Jun 25 2020: (Start)
%F Sum_{n>=0) 1/a(n) = sinh(sqrt(2))/sqrt(2).
%F Sum_{n>=0) (-1)^n/a(n) = sin(sqrt(2))/sqrt(2). (End)
%p a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]*(2*n-1)*(n-1) od: seq(a[n], n=1..14); # _Zerinvary Lajos_, Mar 08 2008
%t Table[(2n+1)!/2^n,{n,0,20}] (* _Harvey P. Dale_, May 13 2011 *)
%o (Magma) [Factorial(2*n+1)/2^n: n in [0..25]]: // _Vincenzo Librandi_, May 14 2011
%o (PARI) a(n) = (2*n+1)!/2^n; \\ _Altug Alkan_, Aug 27 2018
%Y Numerators: A002067, erf(x): A007680.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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