1,3

a(n)=Sum[a1!a2!...ak! ] where (a1,a2,...,ak) ranges over all compositions of n-1. a(n) = number of trees on [0,n-1] rooted at 0, consisting entirely of filaments, and such that the non-root labels on each filament, when arranged in order, form an interval of integers. A filament is a maximal path (directed away from the root) whose interior vertices all have outdegree 1 and which terminates at a leaf. For example with n=4, a(n) = 11 counts all n^(n-2) = 16 trees on [0,3] except the 3 trees {0->1, 1->2, 1->3,}, {0->2, 2->1, 2->3}, {0->3, 3->1, 3->2} (they fail the all-filaments test) and the 2 trees {0->2, 0->3, 3->1}, {0->2, 0->1, 1->3} (they fail the interval-of-integers test). - David Callan (callan(AT)stat.wisc.edu), Oct 24 2004

T(n,k) is the number of lists of "unlabeled" permutations whose total length is n. "Unlabeled" means each permutation is on an initial segment of the positive integers. Example: with dashes separating permutations, a(3) = 11 counts 123, 132, 213, 231, 312, 321, 1-12, 1-21, 12-1, 21-1, 1-1-1. - David Callan (callan(AT)stat.wisc.edu), Sep 20 2007

L. Comtet, Sur les coefficients de l'inverse de la serie formelle Sum n! t^n, Comptes Rendus Acad. Sci. Paris, A 275 (1972), 569-572.

L. Comtet, Advanced Combinatorics, Reidel, 1974.

G.f.: 1/(1-Sum(n!*x^n,n=1..infinity)).

a[ 1 ]=1, a[ m+1 ]=sum {k=1 to m} [ a[ m+1-k ]*k! ].

a(n+1) = Sum_{k>=0} A090238(n, k). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 05 2004

a[ 5 ]=47=a[ 4 ]*1!+a[ 3 ]*2!+a[ 2 ]*3!+a[ 1 ]*4!=11*1+3*2+1*6+1*24

Cf. A051295.

Sequence in context: A062146 A090365 A035009 this_sequence A030832 A030865 A030902

Adjacent sequences: A051293 A051294 A051295 this_sequence A051297 A051298 A051299

easy,nonn

Leroy Quet (qq-quet(AT)mindspring.com)

Entry revised by David Callan (callan(AT)stat.wisc.edu), Sep 20 2007