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A120730 Another version of Catalan triangle A009766. 25
1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 0, 5, 4, 1, 0, 0, 0, 5, 9, 5, 1, 0, 0, 0, 0, 14, 14, 6, 1, 0, 0, 0, 0, 14, 28, 20, 7, 1, 0, 0, 0, 0, 0, 42, 48, 27, 8, 1, 0, 0, 0, 0, 0, 42, 90, 75, 35, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, ...] DELTA [1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -1, 1, ...] where DELTA is the operator defined in A084938.

Aerated version gives A165408. [From Philippe Deléham, Sep 22 2009]

T(n,k) is the number of length n left factors of Dyck paths having k up steps. Example: T(5,4)=4 because we have UDUUU, UUDUU, UUUDU, and UUUUD, where U=(1,1) and D=(1,-1). [Emeric Deutsch, Jun 19 2011]

With zeros omitted : 1,1,1,1,2,1,2,3,1,5,4,1,... = A008313. - From Philippe Deléham, Nov 02 2011

LINKS

Table of n, a(n) for n=0..65.

FORMULA

G.f.: G(t,z)=4*z/((2*z-1+sqrt(1-4*t*z^2))*(1+sqrt(1-4*t*z^2)). [Emeric Deutsch, Jun 19 2011]

Sum_{k, 0<=k<=n}x^k*T(n,n-k)= A001405(n), A126087(n), A128386(n), A121724(n), A128387(n), A132373(n), A132374(n), A132375(n), A121725(n) for x=1,2,3,4,5,6,7,8,9 respectively .

T(2*n,n)= A000108(n); A000108 : Catalan numbers.

Sum_{k, 0<=k<=n}T(n,k)^2 = A000108(n) and Sum_{n, n>=k}T(n,k) = A000108(k+1). [From Philippe Deléham, Oct 18 2008]

Sum_{k, 0<=k<=n}T(n,k)^3 = A003161(n). [From Philippe Deléham, Oct 18 2008]

Sum_{k, 0<=k<=n}T(n,k)^4 = A129123(n). [From Philippe Deléham, Oct 18 2008]

Sum{k=0..n, T(n,k)*x^k}= A000007(n), A001405(n), A151281(n), A151162(n), A151254(n), A156195(n), A156361(n), A156362(n), A156566(n), A156577(n) for x=0,1,2,3,4,5,6,7,8,9 respectively. [From Philippe Deléham, Feb 10 2009]

EXAMPLE

As a triangle, this begins:

1;

0, 1;

0, 1, 1;

0, 0, 2, 1;

0, 0, 2, 3, 1;

0, 0, 0, 5, 4, 1;

0, 0, 0, 5, 9, 5, 1;

0, 0, 0, 0, 14, 14, 6, 1;

MAPLE

G := 4*z/((2*z-1+sqrt(1-4*z^2*t))*(1+sqrt(1-4*z^2*t))): Gser := simplify(series(G, z = 0, 13)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form  # Emeric Deutsch, Jun 19 2011

CROSSREFS

Cf. A008313, A009766, A039598, A039599

Sequence in context: A245842 A191400 A168315 * A122851 A064301 A199881

Adjacent sequences:  A120727 A120728 A120729 * A120731 A120732 A120733

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Aug 17 2006, corrected Sep 15 2006

EXTENSIONS

Corrected formula . - Philippe Deléham, Oct 16 2008

STATUS

approved

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Last modified December 11 21:15 EST 2017. Contains 295919 sequences.