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 A129182 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n such that the area between the x-axis and the path is k (n>=0; 0<=k<=n^2). 9
 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 3, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 6, 0, 7, 0, 7, 0, 5, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 5, 0, 10, 0, 14, 0, 17, 0, 16, 0, 16, 0, 14, 0, 11, 0, 9, 0, 7, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,14 COMMENTS Row n has n^2 + 1 terms. Row sums are the Catalan numbers (A000108). Sum(k*T(n,k), k=0..n^2) = A008549(n). Sums along falling diagonals give A005169. - Joerg Arndt, Mar 29 2014 T(2n,4n) = A240008(n). - Alois P. Heinz, Mar 30 2014 LINKS Alois P. Heinz, Rows n = 0..32, flattened FORMULA G.f.: G(t,z) given by G(t,z) = 1+t*z*G(t,t^2*z)*G(t,z). Sum_{k=0..n^2} (n^2-k)/2 * T(n,k) = A139262(n). - Alois P. Heinz, Mar 31 2018 EXAMPLE T(4,10) = 3 because we have UDUUUDDD, UUUDDDUD and UUDUDUDD. Triangle starts: 1; 0,1; 0,0,1,0,1; 0,0,0,1,0,2,0,1,0,1; 0,0,0,0,1,0,3,0,3,0,3,0,2,0,1,0,1; 0,0,0,0,0,1,0,4,0,6,0,7,0,7,0,5,0,5,0,3,0,2,0,1,0,1; Transposed triangle (A239927) begins: 00: 1; 01: 0, 1; 02: 0, 0, 1; 03: 0, 0, 0, 1; 04: 0, 0, 1, 0, 1; 05: 0, 0, 0, 2, 0, 1; 06: 0, 0, 0, 0, 3, 0, 1; 07: 0, 0, 0, 1, 0, 4, 0, 1; 08: 0, 0, 0, 0, 3, 0, 5, 0, 1; 09: 0, 0, 0, 1, 0, 6, 0, 6, 0, 1; 10: 0, 0, 0, 0, 3, 0, 10, 0, 7, 0, 1; 11: 0, 0, 0, 0, 0, 7, 0, 15, 0, 8, 0, 1; 12: 0, 0, 0, 0, 2, 0, 14, 0, 21, 0, 9, 0, 1; 13: 0, 0, 0, 0, 0, 7, 0, 25, 0, 28, 0, 10, 0, 1; 14: 0, 0, 0, 0, 1, 0, 17, 0, 41, 0, 36, 0, 11, 0, 1; 15: 0, 0, 0, 0, 0, 5, 0, 35, 0, 63, 0, 45, 0, 12, 0, 1; 16: 0, 0, 0, 0, 1, 0, 16, 0, 65, 0, 92, 0, 55, 0, 13, 0, 1; 17: 0, 0, 0, 0, 0, 5, 0, 40, 0, 112, 0, 129, 0, 66, 0, 14, 0, 1; 18: 0, 0, 0, 0, 0, 0, 16, 0, 86, 0, 182, 0, 175, 0, 78, 0, 15, 0, 1; 19: 0, 0, 0, 0, 0, 3, 0, 43, 0, 167, 0, 282, 0, 231, 0, 91, 0, 16, 0, 1; 20: 0, 0, 0, 0, 0, 0, 14, 0, 102, 0, 301, 0, 420, 0, 298, 0, 105, 0, 17, 0, 1; ... - Joerg Arndt, Mar 25 2014 MAPLE G:=1/(1-t*z*g[1]): for i from 1 to 11 do g[i]:=1/(1-t^(2*i+1)*z*g[i+1]) od: g[12]:=0: Gser:=simplify(series(G, z=0, 11)): for n from 0 to 7 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 7 do seq(coeff(P[n], t, j), j=0..n^2) od; # yields sequence in triangular form # second Maple program: b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1, expand(b(x-1, y-1)*z^(y-1/2)+ b(x-1, y+1)*z^(y+1/2)))) end: T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0)): seq(T(n), n=0..10); # Alois P. Heinz, Mar 29 2014 MATHEMATICA b[x_, y_] := b[x, y] = If[y<0 || y>x, 0, If[x==0, 1, Expand[b[x-1, y-1]*z^(y-1/2) + b[x-1, y+1]*z^(y+1/2)]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *) CROSSREFS Cf. A000108, A008549, A139262, A240008, A143951 (column sums). Sequence in context: A323591 A105348 A016406 * A116857 A322338 A158971 Adjacent sequences: A129179 A129180 A129181 * A129183 A129184 A129185 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Apr 08 2007 STATUS approved

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Last modified June 6 05:06 EDT 2023. Contains 363139 sequences. (Running on oeis4.)