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
Keywords
Examples
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.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..300
Programs
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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);
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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 *)
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
Comments