A112308 Sum of the heights of the second peaks in all Dyck paths of semilength n+2.

1, 6, 25, 93, 333, 1180, 4183, 14895, 53349, 192239, 696765, 2539157, 9299547, 34215102, 126411177, 468822297, 1744799967, 6514363557, 24393558687, 91591471287, 344764147407, 1300756937445, 4918188617379, 18633066901747

Offset

0,2

Comments

a(n) = Sum_{k=0..n+1} k*A112307(n+2,k).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

G.f.: c^4*(1+z*c)/(1-z), where c=(1-sqrt(1-4*z))/(2*z) is the Catalan function.

Recurrence: (n+4)*(221*n-49)*a(n) = (1105*n^2 + 2877*n + 1178)*a(n-1) - 2*(442*n^2 + 1077*n + 659)*a(n-2) + 56*(2*n-1)*a(n-3). - Vaclav Kotesovec, Oct 19 2012

D-finite with recurrence 2*(n+4)*a(n) +(-15*n-38)*a(n-1) +2*(17*n+20)*a(n-2) +(-25*n-4)*a(n-3) +2*(2*n-3)*a(n-4)=0. - R. J. Mathar, Jul 26 2022

a(n) ~ 13*2^(2*n+4)/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012

Example

a(1)=6 because the second peaks of the Dyck paths UDUDUD, UDUUDD, UUDDUD, UUDUDD and UUUDDD, where U=(1,1), D=(1,-1), are 1, 2, 1, 2 and 0, respectively.

Maple

c:=(1-sqrt(1-4*z))/2/z: g:=series(c^4*(1+z*c)/(1-z), z=0, 32): 1, seq(coeff(g, z^n), n=1..27);

Mathematica

CoefficientList[Series[((1-Sqrt[1-4*x])/(2*x))^4*(1+x*(1-Sqrt[1-4*x])/(2*x))/(1-x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

Crossrefs

Cf. A112307.

Partial sums of A070857.

Sequence in context: A209241 A369360 A092491 * A034336 A291230 A092184

Adjacent sequences: A112305 A112306 A112307 * A112309 A112310 A112311

Keyword

nonn

Author

Emeric Deutsch, Nov 30 2005

Status

approved