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 A191523 Triangle read by rows: T(n,k) is the number of left factors of Dyck paths of length n and having k double rises, i.e., two consecutive (1,1)-steps (n>=1, 0<=k<=n-1). 1
 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 1, 6, 6, 5, 1, 1, 1, 6, 12, 8, 6, 1, 1, 1, 10, 20, 20, 10, 7, 1, 1, 1, 10, 30, 35, 28, 12, 8, 1, 1, 1, 15, 50, 70, 54, 37, 14, 9, 1, 1, 1, 15, 65, 115, 116, 75, 47, 16, 10, 1, 1, 1, 21, 105, 210, 224, 175, 99, 58, 18, 11, 1, 1, 1, 21, 126, 315, 420, 357, 246, 126, 70, 20, 12, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS Row n contains n entries. Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n). Sum_{k>=0} k*T(n,k) = A191524(n). LINKS FORMULA G.f.: G(t,z)=(z+r+r*z)/(1-t*z*(1+r)) where r=r(t,z) is a solution of z^2*(1+r)*(1+t*r) (the Narayana function with argument z^2). EXAMPLE T(5,2)=4 because we have UD(U[U)U], (UU)D(UU), (U[U)U]DD, and (U[U)U]DU, where U=(1,1) and D=(1,-1) (the double rises are shown between parentheses). Triangle starts:   1;   1, 1;   1, 1, 1;   1, 3, 1, 1;   1, 3, 4, 1, 1;   1, 6, 6, 5, 1, 1; MAPLE eqr := R = z^2*(1+R)*(1+t*R): r := RootOf(eqr, R): G := (z+r+r*z)/(1-t*z*(1+r)): Gser := simplify(series(G, z = 0, 17)): for n to 13 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 13 do seq(coeff(P[n], t, k), k = 0 .. n-1) end do; # yields sequence in triangular form CROSSREFS Cf. A001405, A191524. Sequence in context: A180683 A214635 A166030 * A132890 A295295 A069290 Adjacent sequences:  A191520 A191521 A191522 * A191524 A191525 A191526 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Jun 05 2011 STATUS approved

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Last modified January 23 22:46 EST 2022. Contains 350515 sequences. (Running on oeis4.)