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A191781 Triangle read by rows: T(n,k) is the number of length-n  left factors of Dyck paths having length of the first ascent equal to k (1<=k<=n). 1
1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 1, 6, 6, 4, 2, 1, 1, 10, 10, 7, 4, 2, 1, 1, 20, 20, 14, 8, 4, 2, 1, 1, 35, 35, 25, 15, 8, 4, 2, 1, 1, 70, 70, 50, 30, 16, 8, 4, 2, 1, 1, 126, 126, 91, 56, 31, 16, 8, 4, 2, 1, 1, 252, 252, 182, 112, 62, 32, 16, 8, 4, 2, 1, 1, 462, 462, 336, 210, 119, 63, 32, 16, 8, 4, 2, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Sum of entries in row n is binom(n,floor(n/2))=A001405(n).

T(n,1)=A001405(n-2) (n>=1).

T(n,2)=A001405(n-2) (n>=2).

Sum(k*T(n,k),k=1..n)=A191782(n).

LINKS

Table of n, a(n) for n=1..91.

FORMULA

G.f.: G(t,z) = (1 - z*c + t*z^3*c^2)/((1 - z*c)*(1 - t*z)*(1 - t*z^2)), where c = (1-sqrt(1 - 4*z^2))/(2*z^2).

EXAMPLE

T(5,2)=3 because we have UUDDU, UUDUD, and UUDUU, where U=(1,1) and D=(1,-1).

Triangle starts:

1;

1,1;

1,1,1;

2,2,1,1;

3,3,2,1,1;

6,6,4,2,1,1;

MAPLE

c := ((1-sqrt(1-4*z^2))*1/2)/z^2: G := (1-z*c+t*z^3*c^2)/((1-z*c)*(1-t*z)*(1-t*z^2*c)): 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 = 1 .. n) end do; % yields sequence in triangular form

CROSSREFS

Cf. A001405, A191782

Sequence in context: A255559 A181935 A027358 * A155092 A095133 A126081

Adjacent sequences:  A191778 A191779 A191780 * A191782 A191783 A191784

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Jun 18 2011

STATUS

approved

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Last modified June 24 05:21 EDT 2019. Contains 324318 sequences. (Running on oeis4.)