%I A008544
%S A008544 1,2,10,80,880,12320,209440,4188800,96342400,2504902400,
%T A008544 72642169600,2324549427200,81359229952000,3091650738176000,
%U A008544 126757680265216000,5577337931669504000,262134882788466688000
%N A008544 Triple factorial numbers: product[ k=0..n-1 ] (3*k+2).
%C A008544 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.
%H A008544 T. D. Noe, <a href="b008544.txt">Table of n, a(n) for n=0..100</a>
%H A008544 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               On generalizations of Stirling number triangles</a>, J. Integer Seqs., 
               Vol. 3 (2000), #00.2.4.
%F A008544 E.g.f. (1-3*x)^(-2/3). a(n)= 2*A034000(n) = (3*n-1)(!^3), n >= 1, a(0) 
               := 1.
%F A008544 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
%F A008544 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
%F A008544 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
%F A008544 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-xt)^(-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
%F A008544 Let b(n)=b(n-1)+3; then a(n)=b(n)*a(n-1). - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), 
               Sep 17 2008
%e A008544 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.
%p A008544 f := n->product( (3*k-1),k=0..n);
%t A008544 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
%Y A008544 a(n)= A004747(n+1, 1) (first column of triangle). Cf. A051141.
%Y A008544 Cf. A000165, A001813, A047055, A047657, A084947, A084948, A084949.
%Y A008544 Cf. A049308, A034724.
%Y A008544 Sequence in context: A152600 A048286 A133480 this_sequence A064312 A063902 
               A088351
%Y A008544 Adjacent sequences: A008541 A008542 A008543 this_sequence A008545 A008546 
               A008547
%K A008544 nonn
%O A008544 0,2
%A A008544 Joe Keane (jgk(AT)jgk.org)