1,3

Product of first n triangular numbers. Might be called a triangular factorial number. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 19 2002

Number of ways of transforming n distinguishable objects into n singletons via a sequence of n-1 refinements. Example: a(3)=3 because we have XYZ->X|YZ->X|Y|Z, XYZ->Y|XZ->X|Y|Z and XYZ->Z|XY->X|Y|Z. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 23 2005

Contribution from David Callan (callan(AT)stat.wisc.edu), Aug 27 2009: (Start)

With offset 0, a(n) = number of unordered increasing full binary trees of 2n edges with leaf set {n,n+1,...,2n}, where full binary means each nonleaf vertex has two children, increasing means the vertices are labeled 0,1,2,...,2n and each child is greater than its parent, unordered might as well mean ordered and each pair of sibling vertices is increasing left to right. For example, a(2)=3 counts the trees with edge lists {01,02,13,14}, {01,03,12,14}, {01,04,12,13}.

PROOF. Given such a tree of size n, to produce a tree of size n+1, two new leaves must be added to the leaf n. Choose any two of the leaf set {n+1,...,2n,2n+1,2n+2} for the new leaves and use the rest to replace the old leaves n+1,...,2n, maintaining relative order. Thus each tree of size n yields (n+2)-choose-2 trees of the next size up. Since the ratio a(n+1)/a(n)=(n+2)-choose-2, the result follows by induction.

Without the condition on the leaves, these trees are counted by the reduced tangent numbers A002105. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

O. Frank and K. Svensson, On probability distributions of single-linkage dendrograms, Journal of Statistical Computation and Simulation, 12 (1981), 121-131.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 148.

P. Erdos, R. K. Guy and J. W. Moon, On refining partitions, J. London Math. Soc., 9 (1975), 565-570.

L. Lovasz, Combin. Problems and Exercises, North-Holland, 1979, p. 165.

F. Murtagh, "Counting dendrograms: a survey", Discrete Applied Mathematics, 7 (1984), 191-199.

T. D. Noe, Table of n, a(n) for n=1..50

Index entries for sequences related to factorial numbers

a(n) = a(n-1)*A000217(n).

a(n) = Polygorial(n, 3) = A000142(n)/A000079(n)*A000142(n+1) = n!/2^n*product(i+2, i=0..n-1) = n!/2^n*pochhammer(2, n) = n!^2/2^n*(n+1) = Polygorial(n, 4)/2^n*(n+1) - Daniel Dockery (peritus(AT)gmail.com) Jun 13, 2003

a(n-1) =(-1)^(n+1)/n^2/det(M_n) where M_n is the matrix M_(i, j)=abs(1/i-1/j) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 21 2003

A006472 := n->n!*(n-1)!/2^(n-1);

seq(mul(binomial(k, 2), k =2..n), n=1..18 ); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 07 2007

Cf. A084939, A084940, A084941, A084942, A084943, A084944.

Equals 2^(n-1) A010790(n-1).

Sequence in context: A080687 A111465 A108994 this_sequence A132853 A084879 A141118

Adjacent sequences: A006469 A006470 A006471 this_sequence A006473 A006474 A006475

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).