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A034000 One half of triple factorial numbers. 21
1, 5, 40, 440, 6160, 104720, 2094400, 48171200, 1252451200, 36321084800, 1162274713600, 40679614976000, 1545825369088000, 63378840132608000, 2788668965834752000, 131067441394233344000, 6553372069711667200000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Contribution from Gary W. Adamson, May 17 2010: (Start)

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). (End)

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..375

J.-C. Novelli, 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(3*j+2, j=0..n), 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

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.

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

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Last modified July 9 14:04 EDT 2020. Contains 335543 sequences. (Running on oeis4.)