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A008544 Triple factorial numbers: product_{k=0..n-1} (3*k+2). 61
1, 2, 10, 80, 880, 12320, 209440, 4188800, 96342400, 2504902400, 72642169600, 2324549427200, 81359229952000, 3091650738176000, 126757680265216000, 5577337931669504000, 262134882788466688000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n-1), n>=1, enumerates increasing plane (aka ordered) trees with n vertices (one of them a root labeled 1) where each vertex with out-degree r>=0 comes in r+1 types (like an (r+1)-ary vertex). See the increasing tree comments under A004747. - Wolfdieter Lang, Oct 12 2007.

An example for the case of 3 vertices is shown below. For the enumeration of non-plane trees of this type see A029768. - Peter Bala, Aug 30 2011

a(n) is the product of the positive integers k <= 3*n that have k modulo 3 = 2. - Peter Luschny, Jun 23 2011

See A094638 for connections to differential operators. - Tom Copeland, Sep 20 2011

Partial products of A016789. - Reinhard Zumkeller, Sep 20 2013

The Mathar conjecture is true. Generally from the factorial form, the last term is the "extra" product beyond the prior term, from k=n-1 and 3k+2 evaluates to 3*(n-1)+2 = 3n-1, yielding a(n) = a(n-1)*(3n-1) (eqn1). Similarly, a(n) = a(n-2)*(3n-1)*(3(n-2)+2) = a(n-2)*(3n-1)*(3n-4) (eqn2) and a(n) = a(n-3)*(3n-1)*(3n-4)*(3*(n-2)+2) = a(n-3)*(3n-1)*(3n-4)*(3n-7) (eqn3). We equate (eqn2) and (eqn3) to get a(n-2)*(3n-1)*(3n-4) = a(n-3*(3n-1)*(3n-4)*(3n-7) or a(n-2)+(7-3n)*a(n-3) = 0 (eqn4). From (eqn1) we have a(n)+(1-3n)*a(n-1) = 0 (eqn5). Combining (eqn4) and (eqn5) yields a(n)+(1-3n)*a(n-1)+a(n-2)+(7-3n)*a(n-3) = 0. - Bill McEachen, Jan 01 2016

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

FORMULA

a(n) = Product_{k=0..n-1} (3*k+2) = A007661(3*n-1) (with A007661(-1) = 1).

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

a(n) = 2*A034000(n), n >= 1, a(0) = 1.

a(n) ~ 2^(1/2)*Pi^(1/2)*Gamma(2/3)^-1*n^(1/6)*3^n*e^-n*n^n*{1 - 1/36*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 22 2001

a(n) = (GAMMA(2*n-5/3)/GAMMA(n-5/6)*GAMMA(2/3)/GAMMA(5/6))/sqrt(3)*3^n/4^(n-1). - Jeremy L. Martin, Mar 31 2002 (typo fixed by Vincenzo Librandi, Feb 21 2015)

a(n) = A084939(n)/A000142(n)*A000079(n) = 3^n*pochhammer(2/3, n) = 3^n*GAMMA(n+2/3)/GAMMA(2/3). - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Let T = A094638 and c(t) = column vector(1, t, t^2, t^3, t^4, t^5,...), then A008544 = unsigned [ T * c(-3) ] and the list partition transform A133314 of [1,T * c(-3)] gives [1,T * c(3)] with all odd terms negated, which equals a signed version of A007559; i.e., LPT[(1,signed A008544)] = signed A007559. Also LPT[A007559] = (1,-A008544) and e.g.f. [1,T * c(t)] = (1-x*t)^(-1/t) for t = 3 or -3. Analogous results hold for the double factorial, quadruple factorial and so on. - Tom Copeland, Dec 22 2007

Conjecture: a(n) + (1-3*n)*a(n-1) + a(n-2) + (7-3*n)*a(n-3) = 0. - R. J. Mathar, Nov 14 2011

G.f.: 1/(1-2x/(1-3x/(1-5x/(1-6x/(1-8x/(1-9x/(1-11x/(1-12x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012

a(n) = (-1)^n*Sum_{k=0..n} 3^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+2)/(1 - x*(3*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(3*k+2)/(x*(3*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013

a(n) = (9*(n-2)*(n-1)+2)*a(n-2) + 4*a(n-1), n>=2. - Ivan N. Ianakiev, Aug 09 2013

a(n) = n!*Sum_{k=floor(n/2)..n} binomial(k,n-k)*binomial(n+k,k)*3^(-n+k)*(-1)^(n-k). - Vladimir Kruchinin, Sep 28 2013

Recurrence equation: a(n) = 3*a(n-1) + (3*n - 4)^2*a(n-2) with a(0) = 1 and a(1) = 2. A024396 satisfies the same recurrence (but with different initial conditions). This observation leads to a continued fraction expansion for the constant A193534 due to Euler. - Peter Bala, Feb 20 2015

a(n) = A225470(n, 0), n >= 0. - Wolfdieter Lang, May 29 2017

EXAMPLE

a(2)=10 from the described trees with 3 vertices: there are three trees with a root vertex (label 1) with out-degree r=2 (like the three 3-stars each with one different ray missing) and the four trees with a root (r=1 and label 1) a vertex with (r=1) and a leaf (r=0). Assigning labels 2 and 3 yields 2*3+4=10 such trees.

a(2) = 10. The 10 possible plane increasing trees on 3 vertices, where vertices of out degree 1 come in 2 colors (denoted a or b) and vertices of outdegree 2 come in 3 colors (a,b or c}, are:

.

   1a    1b    1a    1b        1a       1b       1c

   |     |     |     |        / \      / \      / \

   2a    2b    2b    2a      2   3    2   3    2   3

   |     |     |     |

   3     3     3     3         1a       1b       1c

                              / \      / \      / \

                             3   2    3   2    3   2

MAPLE

a := n -> mul(3*k-1, k = 1..n);

A008544 := n -> mul(k, k = select(k-> k mod 3 = 2, [$1 .. 3*n])): seq(A008544(n), n = 0 .. 16); # Peter Luschny, Jun 23 2011

MATHEMATICA

k = 3; b[1] = 2; b[n_] := b[n] = b[n - 1] + k; a[0] = 1; a[1] = 2; a[n_] := a[n] = a[n - 1]*b[n]; Table[a[n], {n, 0, 20}] (* Roger L. Bagula, Sep 17 2008 *)

Product[3 k + 2, {k, 0, # - 1}] & /@ Range[0, 16] (* Michael De Vlieger, Jan 02 2016 *)

PROG

(PARI) a(n) = prod(k=0, n-1, 3*k+2 );

(Sage)

@CachedFunction

def A008544(n): return 1 if n == 0 else (3*n-1)*A008544(n-1)

[A008544(n) for n in (0..16)]  # Peter Luschny, May 20 2013

(Haskell)

a008544 n = a008544_list !! n

a008544_list = scanl (*) 1 a016789_list

-- Reinhard Zumkeller, Sep 20 2013

(Maxima)

a(n):=((n)!*sum(binomial(k, n-k)*binomial(n+k, k)*3^(-n+k)*(-1)^(n-k), k, floor(n/2), n)); /* Vladimir Kruchinin, Sep 28 2013 */

(MAGMA) [Round((Gamma(2*n-5/3)/Gamma(n-5/6)*Gamma(2/3)/Gamma(5/6))/Sqrt(3)*3^n/4^(n-1)): n in [1..20]]; // Vincenzo Librandi, Feb 21 2015

CROSSREFS

a(n) = A004747(n+1, 1) (first column of triangle). Cf. A051141.

Cf. A000165, A001813, A047055, A047657, A084947, A084948, A084949.

Cf. A049308, A034724, A029768.

Cf. Subsequence of A007661. Cf. A024396, A193534.

Cf. A225470, A290596 (first columns).

Sequence in context: A048286 A227463 A133480 * A227464 A269353 A064312

Adjacent sequences:  A008541 A008542 A008543 * A008545 A008546 A008547

KEYWORD

nonn,easy,changed

AUTHOR

Joe Keane (jgk(AT)jgk.org)

STATUS

approved

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Last modified September 24 18:54 EDT 2017. Contains 292433 sequences.