A034000 One half of triple factorial numbers.

1, 5, 40, 440, 6160, 104720, 2094400, 48171200, 1252451200, 36321084800, 1162274713600, 40679614976000, 1545825369088000, 63378840132608000, 2788668965834752000, 131067441394233344000, 6553372069711667200000, 347328719694718361600000, 19450408302904228249600000

Offset: 1

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

Preface the series with a 1, then the next term = (1, 4, 7, 10, ...) dot (1, 1, 5, 40, ...). E.g., a(5) = 6160 = (1, 4, 7, 10, 13) dot (1, 1, 5, 40, 440) = (1 + 4 + 35 + 400 + 5720). - Gary W. Adamson, May 17 2010

In other words, a(n) = Sum_{i=0..n-1} b(i)*A016777(i) where b(0)=1 and b(n)=a(n). - Michel Marcus, Dec 18 2022

Links

• G. C. Greubel, Table of n, a(n) for n = 1..375
• J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.

Crossrefs

Cf. A007559, A034001, A025748, A034724, A016777.

Programs

GAP

a:=[1];; for n in [2..20] do a[n]:=(3*n-1)*a[n-1]; od; a; # G. C. Greubel, Aug 15 2019

Magma

[n le 1 select 1 else (3*n-1)*Self(n-1): n in [1..20]]; // G. C. Greubel, Aug 15 2019

Maple

A034000:=n->`if`(n=1, 1, (3*n-1)*A034000(n-1)); seq(A034000(n), n=1..20); # G. C. Greubel, Aug 15 2019

Mathematica

nxt[{n_,a_}]:={n+1,(3(n+1)-1)*a}; Transpose[NestList[nxt,{1,1},20]][[2]] (* Harvey P. Dale, Aug 22 2015 *)
Table[3^(n-1)*Pochhammer[5/3, n-1], {n,20}] (* G. C. Greubel, Aug 15 2019 *)

PARI

m=20; v=concat([1], vector(m-1)); for(n=2, m, v[n]=(3*n-1)*v[n-1]); v \\ G. C. Greubel, Aug 15 2019

Sage

def a(n):
    if n==1: return 1
    else: return (3*n-1)*a(n-1)
[a(n) for n in (1..20)] # G. C. Greubel, Aug 15 2019

Formula

a(n) = A007661(3n-1)/2 = A008544(n)/2.

2*a(n+1) = (3*n+2)!!! = Product_{j=0..n} (3*j+2), n >= 0.

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

a(n) = (3*n-1)!/(2*3^(n-1)*(n-1)!*A007559(n)).

a(n) ~ 3/2*2^(1/2)*Pi^(1/2)*Gamma(2/3)^-1*n^(7/6)*3^n*e^-n*n^n*{1 + 23/36*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001

a(n) = 3^n*(n+2/3)!/(2/3)!, with offset 0. - Paul Barry, Sep 04 2005

D-finite with recurrence a(n) + (1-3*n)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012

Sum_{n>=1} 1/a(n) = 2*(e/3)^(1/3)*(Gamma(2/3) - Gamma(2/3, 1/3)). - Amiram Eldar, Dec 18 2022