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 A300323 Number of Dyck paths of semilength n such that the area under the right half of the path equals the area under the left half of the path. 3
 1, 1, 2, 3, 6, 12, 28, 69, 186, 522, 1536, 4638, 14408, 45568, 146884, 479871, 1589516, 5320854, 18000198, 61412376, 211282386, 731973720, 2553168136, 8957554412, 31604599044, 112060048354, 399227283950, 1428315878002, 5130964125124, 18499652813682 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..300 Wikipedia, Counting lattice paths FORMULA a(n) >= A001405(n) with equality only for n <= 4. a(n) is odd <=> n in { A000225 }. EXAMPLE /\ / \ /\/\ a(3) = 3: / \ / \ /\/\/\ . . a(5) = 12 counts A001405(5) = 10 symmetric plus 2 non-symmetric Dyck paths: /\ /\ /\/ \/ \ and its reversal. MAPLE b:= proc(x, y) option remember; expand(`if`(x=0, 1, `if`(y<1, 0, b(x-1, y-1)*z^(2*y-1))+ `if`(x add(coeff(p, z, i)^2 , i=0..degree(p)))(b(n, n-2*j)), j=0..n/2) end: seq(a(n), n=0..32); MATHEMATICA b[x_, y_] := b[x, y] = Expand[If[x == 0, 1, If[y < 1, 0, b[x - 1, y - 1] z^(2y - 1)] + If[x < y + 2, 0, b[x - 1, y + 1] z^(2y + 1)]]]; a[n_] := a[n] = Sum[Function[p, Sum[Coefficient[p, z, i]^2, {i, 0, Exponent[p, z]}]][b[n, n - 2j]], {j, 0, n/2}]; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, May 31 2018, from Maple *) CROSSREFS Column k=0 of A300322. Cf. A000108 (all Dyck paths), A000225, A001405 (symmetric Dyck paths), A129182, A239927, A298645. Sequence in context: A337717 A003317 A145062 * A261230 A014278 A061056 Adjacent sequences: A300320 A300321 A300322 * A300324 A300325 A300326 KEYWORD nonn AUTHOR Alois P. Heinz, Mar 02 2018 STATUS approved

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Last modified December 2 23:17 EST 2023. Contains 367526 sequences. (Running on oeis4.)