A208739 2^n minus the number of partitions of n.

0, 1, 2, 5, 11, 25, 53, 113, 234, 482, 982, 1992, 4019, 8091, 16249, 32592, 65305, 130775, 261759, 523798, 1047949, 2096360, 4193302, 8387353, 16775641, 33552474, 67106428, 134214718, 268431738, 536866347, 1073736220, 2147476806, 4294958947, 8589924449

Offset: 0

Author

David Nacin, Mar 01 2012

Keywords

• nonn

Comments

Gives the number of multisets that occurring as the peak heights multiset of a Dyck (n+1)-path minus the number of multisets occurring as the peak heights multiset of a Dyck n-path. We use the definition given by Callan and Deutsch (see reference). A Dyck n-path 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 x-axis. A peak is an occurrence of U D and the peak height is the y-coordinate of the vertex between its U and D.

Examples

			For n=2 the possibilities are UDUD, UUDD giving us multisets of {1,1} and {2} respectively.  For n=1 there is only the one possibility UD giving us {1}.  Thus a(1) = 2 - 1 = 1.
		

Links

• D. Callan and E. Deutsch, Problems and Solutions: 11624, The Amer. Math. Monthly 119 (2012), no. 2, 161-162.

Crossrefs

Cf. A000041, A000079, A208738, A208740.

Programs

Maple

a:= n-> 2^n-combinat[numbpart](n):
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 14 2024

Mathematica

Table[2^n - PartitionsP[n], {n, 0, 40}]

PARI

a(n) = 2^n - numbpart(n); \\ Michel Marcus, Jul 05 2018

Formula

a(n) = A208738(n+1) - A208738(n).

G.f.: 1/(1-2x) - Product_{k>0} 1/(1-x^k).

a(n) = A000079(n) - A000041(n). - Alois P. Heinz, Feb 14 2024

Extensions

Missing a(0)=0 inserted by Alois P. Heinz, Feb 14 2024