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A118973 Number of hill-free Dyck paths of semilength n+2 and having length of first descent equal to 2 (a hill in a Dyck path is a peak at level 1). 3
1, 0, 2, 5, 16, 51, 168, 565, 1934, 6716, 23604, 83806, 300154, 1083137, 3934404, 14374413, 52787766, 194746632, 721435884, 2682522918, 10008240456, 37455101382, 140569122624, 528926230530, 1994980278636, 7541234323096 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also, for a given j>=2, number of hill-free Dyck paths of semilength n+j and having length of first descent equal to j. a(n)=A000108(n+1)-A000108(n)-[A000957(n+2)-A000957(n+1)]. Columns 2,3,4,... of A118972 (without the initial 0's).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

FORMULA

G.f.: (1-x)*C*F, where F = (1-sqrt(1-4*x))/(x*(3-sqrt(1-4*x)) and C = (1-sqrt(1-4*x))/(2*x) is the Catalan function.

a(n) ~ 5*4^n/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

a(n) = (Sum_{k=0..n-1}((k+2)*(-1)^k*(binomial(2*n-k+1,n-k)/(n+2)-binomial(2*n-k-1,n-k-1)/(n+1))))+(-1)^(n). - Vladimir Kruchinin. Mar 06 2016

EXAMPLE

a(2)=2 because we have uu(dd)uudd and uuu(dd)udd, where u=(1,1),d=(1,-1) (the first descents are shown between parentheses).

MAPLE

F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: g:=(1-z)*C*F: gser:=series(g, z=0, 33): seq(coeff(gser, z, n), n=0..28);

MATHEMATICA

CoefficientList[Series[(1-x)*(1-Sqrt[1-4*x])/x/(3-Sqrt[1-4*x])*(1-Sqrt[1-4*x])/2/x, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

PROG

(Maxima)

a(n):=(sum((k+2)*(-1)^k*(binomial(2*n-k+1, n-k)/(n+2)-binomial(2*n-k-1, n-k-1)/(n+1)), k, 0, n-1))+(-1)^(n); /*  Vladimir Kruchinin. Mar 06 2016 */

(PARI) x='x+O('x^25); Vec((1-x)*(1-sqrt(1-4*x))/x/(3-sqrt(1-4*x))*(1-sqrt(1-4*x))/2/x) \\ G. C. Greubel, Feb 08 2017

CROSSREFS

Cf. A000108, A000957, A118972, A001558.

Sequence in context: A148384 A148385 A205501 * A148386 A148387 A121651

Adjacent sequences:  A118970 A118971 A118972 * A118974 A118975 A118976

KEYWORD

nonn

AUTHOR

Emeric Deutsch, May 08 2006

STATUS

approved

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Last modified May 24 11:23 EDT 2019. Contains 323529 sequences. (Running on oeis4.)