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A355481 Number of pairs of Dyck paths of semilength n such that the midpoint of the first is above the midpoint of the second. 3
0, 0, 1, 4, 49, 441, 4806, 52956, 614713, 7341697, 90118054, 1130414649, 14447230854, 187609607862, 2470253990556, 32922380442828, 443493622670313, 6031353319151961, 82725531355436886, 1143385727109903585, 15913217995801644870, 222875331740976566070 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
FORMULA
a(n) = (A001246(n) - A129123(n))/2 = (A000108(n)^2 - A129123(n))/2.
MAPLE
b:= proc(n) option remember; `if`(n<2, 1, (2*n*(90*n^5-309*n^4+147*n^3+
124*n^2-135*n+35)*b(n-1)+4*(n-1)^2*(4*n-5)*(4*n-3)*(15*n^2-4*n-12)*
b(n-2))/(n*(n+1)^3*(15*n^2-34*n+7)))
end:
a:= n-> ((binomial(n+n, n)/(n+1))^2-b(n))/2:
seq(a(n), n=0..21);
MATHEMATICA
A129123[n_] := Sum[(Binomial[n, k]-Binomial[n, k-1])^4, {k, 0, Floor[n/2]}];
a[n_] := (CatalanNumber[n]^2 - A129123[n])/2;
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 16 2022 *)
CROSSREFS
Sequence in context: A045787 A067474 A053769 * A362479 A173038 A198971
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Oct 07 2022
STATUS
approved

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Last modified February 22 00:43 EST 2024. Contains 370239 sequences. (Running on oeis4.)