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A258174
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Sum over all Dyck paths of semilength n of products over all peaks p of x_p*y_p, where x_p and y_p are the coordinates of peak p.
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10
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1, 1, 7, 84, 1486, 35753, 1111931, 43150593, 2035666985, 114412223081, 7538224510181, 574552299138202, 50096579094908148, 4949493445607316419, 549534510282406667069, 68071071679372210762156, 9347203754680124767253730, 1414740620049957735248175695
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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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Alois P. Heinz, Table of n, a(n) for n = 0..250
Wikipedia, Lattice path
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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*y, 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*y, 1] + b[x - 1, y + 1, True]]];
a[n_] := b[2*n, 0, False];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)
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CROSSREFS
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Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258175, A258176, A258177, A258178, A258179, A258180, A258181.
Sequence in context: A341966 A034323 A172455 * A254569 A183177 A058795
Adjacent sequences: A258171 A258172 A258173 * A258175 A258176 A258177
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KEYWORD
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nonn
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AUTHOR
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Alois P. Heinz, May 22 2015
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STATUS
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approved
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