1,1

Consider the set of all pairs of nonintersecting Dyck excursions of length 2*n (nonnegative walks with jumps -1,+1). The lower path begins and ends at 0; the upper path begins and ends at 2. a(n) is the sum of heights of all such upper-Dyck excursions.

M. Fulmek, Asymptotics of the average height of 2-watermelons with a wall, Elec. J. Combin. 14 (2007) R64

c[n_] := 6*(2*n)!*(2*n+2)!/(n!*(n+1)!*(n+2)!*(n+3)!) s[n_, a_] := Sum[If[k < 1, 0, DivisorSigma[0, k]*Binomial[2*n, n+a-k]/Binomial[2*n, n]], {k, a-n, a+n}] t[n_, a_, b_] := Sum[If[(j < 1) || (k < 1), 0, DivisorSigma[0, GCD[j, k]]*Binomial[2*n, n+a-j]*Binomial[2*n, n+b-k]/Binomial[2*n, n]^2], {j, a-n, a+n}, {k, b-n, b+n}] f[n_] := (n^2+5*n+6)*(s[n, -3]+s[n, 3])-(6*n^2+18*n)*(s[n, -2]+s[n, 2])+(15*n^2+27*n+6)*(s[n, -1]+s[n, 1])-(20*n^2+28*n+24)*s[n, 0] g[n_] := t[n, -2, -2]-t[n, -1, -3]-2*t[n, -1, -2]+t[n, -1, -1]+2*t[n, -1, 0]-t[n, -1, 3]+2*t[n, 0, -3]-4*t[n, 0, 0]+2*t[n, 0, 3]-t[n, 1, -3]-2*t[n, 1, -2]+2*t[n, 1, -1]+2*t[n, 1, 0]+t[n, 1, 1]-t[n, 1, 3]+2*t[n, 2, -2]-2*t[n, 2, -1]-2*t[n, 2, 1]+t[n, 2, 2] h[n_] := ((n+1)*(n+2)/(12*(2*n+1)))*( (n+1)*(n+2)*(n+3)*g[n]+f[n] ) - 1 a[n_] := h[n]*c[n]

Cf. A005700, A078920.

Sequence in context: A152796 A007807 A075201 this_sequence A007146 A076475 A125556

Adjacent sequences: A136437 A136438 A136439 this_sequence A136441 A136442 A136443

nonn

S. R. Finch (Steven.Finch(AT)inria.fr), Apr 02 2008