a(n) = Product_{k=0..n-1} (3*k + 1).
a(n) = (3*n - 2)!!!.
E.g.f.: (1-3*x)^(-1/3).
a(n) ~ sqrt(2*Pi)/Gamma(1/3)*n^(-1/6)*(3*n/e)^n*(1 - (1/36)/n - ...). - Joe Keane (jgk(AT)jgk.org), Nov 22 2001
a(n) = 3^n*Pochhammer(1/3, n).
a(n) = n!*(1+Sum_{m=1..n} (m/n)*Sum_{k=1..n-m} binomial(k, n-m-k)*(-1/3)^(n-m-k)*binomial(k+n-1, n-1)), n>1. -
Vladimir Kruchinin, Aug 09 2010
a(n) = upper left term in M^n, M = a variant of Pascal (1,3) triangle (Cf.
A095660); as an infinite square production matrix:
1, 3, 0, 0, 0,...
1, 4, 3, 0, 0,...
1, 5, 7, 3, 0,...
...
a(n+1) = sum of top row terms of M^n. (End)
a(n) = (-2)^n*Sum_{k=0..n} (3/2)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind,
A048994. -
Mircea Merca, May 03 2012
G.f.: 1/Q(0) where Q(k) = 1 - x*(3*k+1)/( 1 - x*(3*k+3)/Q(k+1) ); (continued fraction). -
Sergei N. Gladkovskii, Mar 21 2013
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(3*k+1)/(x*(3*k+1) + 1/G(k+1))); (continued fraction). -
Sergei N. Gladkovskii, May 26 2013
E.g.f.: E(0)/2, where E(k)= 1 + 1/(1 - x*(3*k+1)/(x*(3*k+1) + (k+1)/E(k+1))); (continued fraction). -
Sergei N. Gladkovskii, May 26 2013
Let D(x) = 1/sqrt(1 - 2*x) be the e.g.f. for the sequence of double factorial numbers
A001147. Then the e.g.f. A(x) for the triple factorial numbers satisfies D( Integral_{t=0..x} A(t) dt ) = A(x). Cf.
A007696 and
A008548. -
Peter Bala, Jan 02 2015
O.g.f.: hypergeom([1, 1/3], [], 3*x). -
Peter Luschny, Oct 08 2015
a(n) = sigma[3,1]^{(n)}_n, n >= 0, with the elementary symmetric function of degree n in the n numbers 1, 4, 7, ..., 1+3*(n-1), with sigma[3,1]^{(n)}_0 := 1. See the first formula. -
Wolfdieter Lang, May 29 2017
a(n) = (-1)^n /
A008544(n), 0 = a(n)*(+3*a(n+1) -a(n+2)) +a(n+1)*a(n+1) for all n in Z. -
Michael Somos, Sep 30 2018
D-finite with recurrence: a(n) +(-3*n+2)*a(n-1)=0, n>=1. -
R. J. Mathar, Feb 14 2020
Sum_{n>=1} 1/a(n) = (e/9)^(1/3) * (Gamma(1/3) - Gamma(1/3, 1/3)). -
Amiram Eldar, Jun 29 2020
