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A136440 Sum of heights of all 2-watermelons with wall of length 2*n.

%I A136440 #9 Feb 07 2024 09:01:25
%S A136440 3,11,60,406,3171,27411,255617,2528613,26224097,282706396,3147801820,
%T A136440 36022733951,422047425238,5046771514478,61438059222438,
%U A136440 759851375725606,9530872096367508,121063493728881999,1555352365759798758
%N A136440 Sum of heights of all 2-watermelons with wall of length 2*n.
%C A136440 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.
%H A136440 M. Fulmek, <a href="https://doi.org/10.37236/982">Asymptotics of the average height of 2-watermelons with a wall</a>, Elec. J. Combin. 14 (2007) R64.
%t A136440 c[n_] := 6*(2*n)!*(2*n+2)!/(n!*(n+1)!*(n+2)!*(n+3)!)
%t A136440 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 A136440 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}]
%t A136440 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]
%t A136440 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]
%t A136440 h[n_] := ((n+1)*(n+2)/(12*(2*n+1)))*( (n+1)*(n+2)*(n+3)*g[n]+f[n] ) - 1
%t A136440 a[n_] := h[n]*c[n]
%Y A136440 Cf. A005700, A078920.
%K A136440 nonn
%O A136440 1,1
%A A136440 _Steven Finch_, Apr 02 2008