0,2

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.

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.

E.g.f. (1-3*x)^(-2/3). a(n)= 2*A034000(n) = (3*n-1)(!^3), 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-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

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

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.

f := n->product( (3*k-1), k=0..n);

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

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

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

Cf. A049308, A034724.

Adjacent sequences: A008541 A008542 A008543 this_sequence A008545 A008546 A008547

Sequence in context: A152600 A048286 A133480 this_sequence A064312 A063902 A088351

nonn

Joe Keane (jgk(AT)jgk.org)