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 A300322 Number T(n,k) of Dyck paths of semilength n such that 2*k is the difference between the area under the right half of the path and the area under the left half of the path; triangle T(n,k), n>=0, -floor(n*(n-1)/6) <= k <= floor(n*(n-1)/6), read by rows. 3
 1, 1, 2, 1, 3, 1, 1, 3, 6, 3, 1, 2, 5, 8, 12, 8, 5, 2, 1, 4, 9, 16, 22, 28, 22, 16, 9, 4, 1, 1, 4, 11, 21, 34, 49, 60, 69, 60, 49, 34, 21, 11, 4, 1, 2, 7, 15, 31, 53, 82, 114, 147, 171, 186, 171, 147, 114, 82, 53, 31, 15, 7, 2, 1, 5, 13, 30, 56, 95, 150, 216, 293, 371, 445, 495, 522, 495, 445, 371, 293, 216, 150, 95, 56, 30, 13, 5, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Alois P. Heinz, Rows n = 0..60, flattened Wikipedia, Counting lattice paths FORMULA T(n,k) = T(n,-k). T(n,A130518(n)) = A177702(n). EXAMPLE /\ T(3,-1) = 1: / \/\ . /\ / \ /\/\ T(3,0) = 3: / \ / \ /\/\/\ . /\ T(3,1) = 1: /\/ \ . Triangle T(n,k) begins: : 1 ; : 1 ; : 2 ; : 1, 3, 1 ; : 1, 3, 6, 3, 1 ; : 2, 5, 8, 12, 8, 5, 2 ; : 1, 4, 9, 16, 22, 28, 22, 16, 9, 4, 1 ; : 1, 4, 11, 21, 34, 49, 60, 69, 60, 49, 34, 21, 11, 4, 1 ; MAPLE b:= proc(x, y, v) option remember; expand( `if`(min(y, v, x-max(y, v))<0, 0, `if`(x=0, 1, (l-> add(add( b(x-1, y+i, v+j)*z^((y-v)/2+(i-j)/4), i=l), j=l))([-1, 1])))) end: T:= n-> (p-> seq(coeff(p, z, i), i=ldegree(p)..degree(p)))( add(b(n, (n-2*j)\$2), j=0..n/2)): seq(T(n), n=0..12); MATHEMATICA b[x_, y_, v_] := b[x, y, v] = Expand[If[Min[y, v, x - Max[y, v]] < 0, 0, If[x == 0, 1, Function[l, Sum[Sum[b[x - 1, y + i, v + j] z^((y - v)/2 + (i - j)/4), {i, l}], {j, l}]][{-1, 1}]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, Range[Exponent[p, z, Reverse @@ # &], Exponent[p, z]]}]][Sum[b[n, n-2j, n-2j], {j, 0, n/2}]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 31 2018, from Maple *) CROSSREFS Row sums give A000108. Column k=0 gives A300323. Cf. A130518, A129182, A177702, A239927, A298645, A300953. Sequence in context: A279945 A342724 A347046 * A144220 A156826 A130296 Adjacent sequences: A300319 A300320 A300321 * A300323 A300324 A300325 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, Mar 02 2018 STATUS approved

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Last modified December 7 23:30 EST 2023. Contains 367662 sequences. (Running on oeis4.)