cp's OEIS Frontend

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

%I A112308 #16 Jul 26 2022 12:14:10
%S A112308 1,6,25,93,333,1180,4183,14895,53349,192239,696765,2539157,9299547,
%T A112308 34215102,126411177,468822297,1744799967,6514363557,24393558687,
%U A112308 91591471287,344764147407,1300756937445,4918188617379,18633066901747
%N A112308 Sum of the heights of the second peaks in all Dyck paths of semilength n+2.
%C A112308 a(n) = Sum_{k=0..n+1} k*A112307(n+2,k).
%H A112308 Vincenzo Librandi, <a href="/A112308/b112308.txt">Table of n, a(n) for n = 0..300</a>
%F A112308 G.f.: c^4*(1+z*c)/(1-z), where c=(1-sqrt(1-4*z))/(2*z) is the Catalan function.
%F A112308 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
%F A112308 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
%F A112308 a(n) ~ 13*2^(2*n+4)/(3*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 19 2012
%e A112308 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.
%p A112308 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);
%t A112308 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 *)
%Y A112308 Cf. A112307.
%Y A112308 Partial sums of A070857.
%K A112308 nonn
%O A112308 0,2
%A A112308 _Emeric Deutsch_, Nov 30 2005