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A011781 Sextuple factorial numbers: a(n) = Product_{k=0..n-1} (6*k+3). 10
1, 3, 27, 405, 8505, 229635, 7577955, 295540245, 13299311025, 678264862275, 38661097149675, 2435649120429525, 168059789309637225, 12604484198222791875, 1020963220056046141875, 88823800144876014343125, 8260613413473469333910625, 817800727933873464057151875 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Total number of Eulerian circuits in rooted labeled multigraphs with n edges. - Valery A. Liskovets, Apr 07 2002

Number of self-avoiding planar walks starting at (0,0), ending at (n,0), remaining in the east quadrant {(x,y): x >= |y|} and using steps (0,1), (0,-1), (1,1), (-1,-1), and (1,0). - Alois P. Heinz, Oct 13 2016

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Fatemeh Bagherzadeh, M Bremner, S Madariaga, Jordan Trialgebras and Post-Jordan Algebras, arXiv preprint arXiv:1611.01214, 2016

Murray Bremner, Martin Markl, Distributive laws between the Three Graces, arXiv:1809.08191 [math.AT], 2018.

B. Lass, Démonstration combinatoire de la formule de Harer-Zagier, (A combinatorial proof of the Harer-Zagier formula) C. R. Acad. Sci. Paris, Serie I, 333 (2001) No 3, 155-160.

B. Lass, Démonstration combinatoire de la formule de Harer-Zagier, (A combinatorial proof of the Harer-Zagier formula) C. R. Acad. Sci. Paris, Serie I, 333 (2001) No 3, 155-160.

Valery Liskovets, A Note on the Total Number of Double Eulerian Circuits in Multigraphs , Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.5

FORMULA

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

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

G.f.: 1/(1-3x/(1-6x/(1-9x/(1-12x/(1-15x/(1-18x/(1-21x/(1-24x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012

a(n) = (-3)^n*sum_{k=0..n} 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.: T(0), where T(k) = 1 - 3*x*(k+1)/( 3*x*(k+1) - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 24 2013

a(n) = 6^n * gamma(n + 1/2) / sqrt(Pi). - Daniel Suteu, Jan 06 2017

0 = a(n)*(+6*a(n+1) - a(n+2)) + a(n+1)*(+a(n+1)) and a(n) = (-1)^n / a(-n) for all n in Z. - Michael Somos, Jan 06 2017

a(n) +3*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Jan 20 2018

EXAMPLE

G.f. = 1 + 3*x + 27*x^2 + 405*x^3 + 8505*x^4 + 229635*x^5 + 7577955*x^6 + ...

MATHEMATICA

s=1; lst={s}; Do[s+=n*s; AppendTo[lst, s], {n, 2, 5!, 6}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)

Table[Product[6k+3, {k, 0, n-1}], {n, 0, 20}] (* or *) Table[6^(n-1) Pochhammer[ 1/2, n-1], {n, 21}] (* Harvey P. Dale, May 09 2012 *)

PROG

(PARI) {a(n) = if( n<0, (-1)^n / a(-n), (3/2)^n * (2*n)! / n!)}; /* Michael Somos, Feb 10 2002, revised and extended Michael Somos, Jan 06 2017 */

(MAGMA) [(3/2)^n*Factorial(2*n)/Factorial(n):n in [0..20]]; // Vincenzo Librandi, May 09 2012

CROSSREFS

Cf. A001147, A047657, A049308.

Cf. A069736, A277358.

Sequence in context: A138436 A141057 A201696 * A094577 A221624 A108525

Adjacent sequences:  A011778 A011779 A011780 * A011782 A011783 A011784

KEYWORD

nonn,easy

AUTHOR

Lee D. Killough (killough(AT)wagner.convex.com)

STATUS

approved

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Last modified July 17 08:42 EDT 2019. Contains 325098 sequences. (Running on oeis4.)