A034723 a(n) is the n-th sextic factorial number divided by 3.

1, 9, 135, 2835, 76545, 2525985, 98513415, 4433103675, 226088287425, 12887032383225, 811883040143175, 56019929769879075, 4201494732740930625, 340321073352015380625, 29607933381625338114375, 2753537804491156444636875, 272600242644624488019050625

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..345

Formula

3*a(n) = (6*n-3)(!^6) = Product_{j=1..n} (6*j-3) = 3^n*A001147(n) = 3^n*(2*n)!/(2^n*n!).

E.g.f.: (-1 + (1-6*x)^(-1/2))/3.

a(n) = 2*(3/2)^(n-1)*(n+1)!*C(n), where C(n) = A000108(n). - G. C. Greubel, Nov 11 2019

D-finite with recurrence: a(n) + 3*(-2*n+1)*a(n-1) = 0. - R. J. Mathar, Feb 24 2020

From Amiram Eldar, Jan 08 2022: (Start)

Sum_{n>=1} 1/a(n) = e^(1/6)*sqrt(3*Pi/2)*erf(1/sqrt(6)), where erf is the error function.

Sum_{n>=1} (-1)^(n+1)/a(n) = e^(-1/6)*sqrt(3*Pi/2)*erfi(1/sqrt(6)), where erfi is the imaginary error function. (End)

Maple

seq(3^(n-1)*(2*n)!/(2^n*n!), n=1..20); # G. C. Greubel, Nov 11 2019

Mathematica

Table[3^(n-1)*(2*n)!/(2^n*n!), {n, 20}] (* G. C. Greubel, Nov 11 2019 *)

Prog

(PARI) a(n) = prod(j=1, n, 6*j-3)/3; \\ Michel Marcus, Mar 13 2019
(Magma) F:=Factorial; [3^(n-1)*F(2*n)/(2^n*F(n)): n in [1..20]]; // G. C. Greubel, Nov 11 2019
(Sage) f=factorial; [3^(n-1)*f(2*n)/(2^n*f(n)) for n in (1..20)] # G. C. Greubel, Nov 11 2019
(GAP) F:=Factorial;; List([1..20], n-> 3^(n-1)*F(2*n)/(2^n*F(n))); # G. C. Greubel, Nov 11 2019

Crossrefs

Cf. A000108, A001147, A008542, A034689.

Sequence in context: A254282 A235339 A306848 * A188685 A052137 A003376

Adjacent sequences: A034720 A034721 A034722 * A034724 A034725 A034726

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved