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

1,2

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.

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

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 *)

Prog

(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
(Magma) [n le 1 select 1 else (3*n-1)*Self(n-1): n in [1..20]]; // 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
(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

Crossrefs

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

Sequence in context: A258172 A304866 A202477 * A000359 A121886 A282190

Adjacent sequences: A033997 A033998 A033999 * A034001 A034002 A034003

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved