A007019 a(n) = (2n+1)! / 2^n. (Formerly M3126)

1, 3, 30, 630, 22680, 1247400, 97297200, 10216206000, 1389404016000, 237588086736000, 49893498214560000, 12623055048283680000, 3786916514485104000000, 1329207696584271504000000, 539658324813214230624000000, 250941121038144617240160000000, 132496911908140357902804480000000

Offset

0,2

Comments

Denominators of coefficients of the Taylor series of sinh(sqrt(2*x))/(sqrt(2*x)). - J. Zurita (jrzurita(AT)inaoep.mx), Dec 01 2007

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..90

L. Carlitz, The inverse of the error function, Pacific J. Math., 13 (1963), 459-470.

R. Flórez and L. Junes, A relation between triangular numbers and prime numbers, Integers 12 (2012), no. 1, 83-96.

Z. Usiskin, Letter to N. J. A. Sloane, Oct. 1991

Eric Weisstein's World of Mathematics, Inverse Erf

Formula

sin(x)*cosh(x) = Sum_{n>=0} (-1)^floor(n/2)*x^(2n+1)/a(n). - Benoit Cloitre, Feb 02 2002

a(n) = Product_{k=0..n-1} (A000217(n+1) - A000217(k)). - Anton Zakharov, Sep 14 2016

a(n) ~ sqrt(Pi)*2^(n+2)*n^(2*n+3/2)/exp(2*n). - Ilya Gutkovskiy, Sep 14 2016

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

From Amiram Eldar, Jun 25 2020: (Start)

Sum_{n>=0) 1/a(n) = sinh(sqrt(2))/sqrt(2).

Sum_{n>=0) (-1)^n/a(n) = sin(sqrt(2))/sqrt(2). (End)

Maple

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

Mathematica

Table[(2n+1)!/2^n, {n, 0, 20}] (* Harvey P. Dale, May 13 2011 *)

Prog

(Magma) [Factorial(2*n+1)/2^n: n in [0..25]]: // Vincenzo Librandi, May 14 2011
(PARI) a(n) = (2*n+1)!/2^n; \\ Altug Alkan, Aug 27 2018

Crossrefs

Numerators: A002067, erf(x): A007680.

Sequence in context: A184900 A274797 A092677 * A327021 A186941 A082879

Adjacent sequences: A007016 A007017 A007018 * A007020 A007021 A007022

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved