

A208740


Number of multisets that occurring as the peak heights multiset of a Dyck npath that are the also the peak heights multiset of a smaller Dyck path.


2



0, 0, 0, 1, 4, 13, 34, 83, 189, 415, 885, 1853, 3824, 7819, 15876, 32084, 64621, 129860, 260547, 522201, 1045862, 2093646, 4189796, 8382845, 16769878, 33545136, 67097132, 134202986, 268416996, 536847887, 1073713195, 2147448177, 4294923476, 8589880629
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OFFSET

1,5


COMMENTS

We use the definition given by Callan and Deutsch (see reference). A Dyck npath is a lattice path of n upsteps U (changing by (1,1)) and n downsteps D (changing by (1,1)) that starts at the origin and never goes below the xaxis. A peak is an occurrence of U D and the peak height is the ycoordinate of the vertex between its U and D.
Also the number of nonempty multisets S of positive integers satisfying max(S) + S <= n <= sum(S).


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
D. Callan and E. Deutsch, Problems and Solutions: 11624, The Amer. Math. Monthly 119 (2012), no. 2, 161162.


FORMULA

a(n) = 2^(n1)  A000070(n1).
a(n) = A208738(n)  2^(n1).
G.f.: x/(12*x)(x/(1x))*product(m>=1, 1/(1x^m)).


EXAMPLE

For a Dyck 4path there is only one peak heights multiset occurring also for a Dyck 3path. This is {2,2} and occurs for both UUDDUUDD when n=4 and UUDUDD when n=3.


MATHEMATICA

Table[2^(n  1)  Sum[PartitionsP[k], {k, 0, n  1}], {n, 1, 40}]


PROG

(PARI) a(n) = 2^(n1)  sum(k=0, n1, numbpart(k)); \\ Michel Marcus, Jul 07 2018


CROSSREFS

Cf. A000070, A208738, A208739.
Sequence in context: A262200 A213578 A212149 * A127981 A296303 A089453
Adjacent sequences: A208737 A208738 A208739 * A208741 A208742 A208743


KEYWORD

nonn


AUTHOR

David Nacin, Mar 01 2012


STATUS

approved



