0,4

A fountain is formed by starting with a row of coins, then stacking additional coins on top so that each new coin touches two in the previous row.

a(n)=number of Dyck paths for which the sum of the heights of the vertices that terminate an upstep (i.e. peaks and doublerises) is n. Example: a(4)=3 beacuse we have UDUUDD, UUDDUD and UDUDUDUD. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 22 2008

a(n)=number of ordered trees with path length n (follows from previous comment via a standard bijection). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 22 2008

Row sums of A047998. Column sums of A138158. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 22 2008

Probably first studied by Jim Propp (unpublished).

Number of compositions of n with a(1) = 1 and a(i+1) <= a(i) + 1. (Slide each row right 1/2 step relative to the row below, and count the columns.) [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Nov 24 2009]

Glasser, M. L.; Privman, V.; Svrakic, N. M.; Temperley's triangular lattice compact cluster model: exact solution in terms of the $q$ series. J. Phys. A 20 (1987), no. 18, L1275-L1280.

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.

A. M. Odlyzko and H. S. Wilf, The editor's corner: n coins in a fountain, Amer. Math. Monthly, 95 (1988), 840-843.

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

T. D. Noe, Table of n, a(n) for n=0..500

P. Flajolet, Combinatorial aspects of continued fractions, Discrete Mathematics 32 (1980), pp. 125-161.

A. M. Odlyzko, Asymptotic enumeration methods, pp. 1063-1229 of R. L. Graham et al., eds., Handbook of Combinatorics, 1995; see Example 10.7 (pdf, ps)

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 331

A005169(n) = f(n, 1), where f(n, p) = 0 if p > n, 1 if p = n, Sum(1 <= q <= p+1; f(n-p, q)) if p < n.

G.f.=F(t)=Sum(P[k],k=0..infinity), where P[0]=1, P[n]=t*sum(P[j]*P[n-j-1]*t^(n-j-1),j= 0..n-1) for n>=1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 22 2008

An example of a fountain with 19 coins:

... O . O O

.. O O O O O O . O

. O O O O O O O O O

P[0]:=1: for n to 40 do P[n]:=sort(expand(t*(sum(P[j]*P[n-j-1]*t^(n-j-1), j= 0..n-1)))) end do: F:=sort(sum(P[k], k=0..40)): seq(coeff(F, t, j), j=0..36); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 22 2008

Cf. A047998, A138158.

Sequence in context: A096816 A018157 A003065 this_sequence A129852 A065954 A067847

First column of A168396. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Nov 24 2009]

Adjacent sequences: A005166 A005167 A005168 this_sequence A005170 A005171 A005172

nonn,easy,nice,new

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

More terms from David W. Wilson (davidwwilson(AT)comcast.net), Apr 30 2001