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A112308 Sum of the heights of the second peaks in all Dyck paths of semilength n+2. 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n)=sum(k*A112307(n+2,k), k=0..n+1).

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

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.

Sequence in context: A143815 A209241 A092491 * A034336 A291230 A092184

Adjacent sequences:  A112305 A112306 A112307 * A112309 A112310 A112311

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Nov 30 2005

STATUS

approved

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Last modified May 25 02:01 EDT 2020. Contains 334581 sequences. (Running on oeis4.)