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A258178
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Sum over all Dyck paths of semilength n of products over all peaks p of x_p^2, where x_p is the x-coordinate of peak p.
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10
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1, 1, 13, 414, 24324, 2279209, 311524201, 58467947511, 14424374692879, 4525566110365523, 1759527523008436279, 830255082140922306224, 467382831980334193769718, 309419146352957449765072455, 237980526477430552734199922151, 210427994109788912088395561755374
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OFFSET
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0,3
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COMMENTS
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A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.
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LINKS
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MAPLE
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b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
`if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x^2, 1) +
b(x-1, y+1, true) ))
end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);
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MATHEMATICA
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b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, x^2, 1] + b[x - 1, y + 1, True] ]];
a[n_] := b[2*n, 0, False];
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CROSSREFS
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Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258177, A258179, A258180, A258181.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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