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A316730
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Number of permutations of {0,1,...,2n+2} with first element n whose sequence of ascents and descents forms a Dyck path.
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2
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1, 7, 121, 4411, 283073, 28318137, 4076415425, 798519164779, 204292676593353, 66150225395814649, 26444888796754193841, 12792566645739144488693, 7364969554345555373419625, 4976538708651698959601499559, 3900052284443403730374391636689
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OFFSET
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0,2
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COMMENTS
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All terms are odd.
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..224
Wikipedia, Counting lattice paths
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FORMULA
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a(n) = A316728(n+1,n).
a(n) ~ c * 4^n * (n!)^2, where c = 1.897642067924382577976619913635026612792069869805703855808680498665... - Vaclav Kotesovec, Jul 15 2018
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EXAMPLE
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a(0) = 1: 021.
a(1) = 7: 12043, 12430, 13042, 13240, 13420, 14032, 14230.
a(2) = 121: 2301654, 2304165, 2304651, 2305164, ..., 2635041, 2635140, 2645031, 2645130.
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MAPLE
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b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
`if`(t>0, add(b(u-j, o+j-1, t-1), j=1..u), 0)+
`if`(o+u>t, add(b(u+j-1, o-j, t+1), j=1..o), 0))
end:
a:= n-> b(n, n+2, 0):
seq(a(n), n=0..20);
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MATHEMATICA
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b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
If[t > 0, Sum[b[u - j, o + j - 1, t - 1], {j, 1, u}], 0] +
If[o + u > t, Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}], 0]];
a[n_] := b[n, n+2, 0];
a /@ Range[0, 20] (* Jean-François Alcover, Mar 27 2021, after Alois P. Heinz *)
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CROSSREFS
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Cf. A316728.
Sequence in context: A279235 A094088 A012043 * A211103 A012103 A012086
Adjacent sequences: A316727 A316728 A316729 * A316731 A316732 A316733
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KEYWORD
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nonn
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AUTHOR
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Alois P. Heinz, Jul 11 2018
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STATUS
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approved
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