|
| |
|
|
A008544
|
|
Triple factorial numbers: product(k=0..n-1, 3*k+2 ).
|
|
46
| |
|
|
1, 2, 10, 80, 880, 12320, 209440, 4188800, 96342400, 2504902400, 72642169600, 2324549427200, 81359229952000, 3091650738176000, 126757680265216000, 5577337931669504000, 262134882788466688000
(list; graph; refs; listen; history; 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. W. 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
|
|
|
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) = prod(k=0..n-1, 3*k+2 ).
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 Martin (jmartin(AT)math.ucsd.edu), Mar 31 2002
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 (tcjpn(AT)msn.com), 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). - DELEHAM Philippe, Jan 08 2012
|
|
|
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 (rlbagulatftn(AT)yahoo.com), Sep 17 2008
|
|
|
PROG
| (PARI) a(n) = prod(k=0, n-1, 3*k+2 );
|
|
|
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.
Sequence in context: A152600 A048286 A133480 * A064312 A063902 A088351
Adjacent sequences: A008541 A008542 A008543 * A008545 A008546 A008547
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Joe Keane (jgk(AT)jgk.org)
|
| |
|
|
