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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Crossrefs

Numerators: A002067, erf(x): A007680.

Programs

Magma

[Factorial(2*n+1)/2^n: n in [0..25]]; // Vincenzo Librandi, May 14 2011

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 *)

PARI

a(n) = (2*n+1)!/2^n; \\ Altug Alkan, Aug 27 2018

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)