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 A120730 Another version of Catalan triangle A009766. 25

%I

%S 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,

%T 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,

%U 75,35,9,1

%N Another version of Catalan triangle A009766.

%C 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.

%C Aerated version gives A165408. [From _Philippe Deléham_, Sep 22 2009]

%C 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]

%C With zeros omitted : 1,1,1,1,2,1,2,3,1,5,4,1,... = A008313. - From _Philippe Deléham_, Nov 02 2011

%F 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]

%F 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 .

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

%F 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]

%F Sum_{k, 0<=k<=n}T(n,k)^3 = A003161(n). [From _Philippe Deléham_, Oct 18 2008]

%F Sum_{k, 0<=k<=n}T(n,k)^4 = A129123(n). [From _Philippe Deléham_, Oct 18 2008]

%F 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]

%e As a triangle, this begins:

%e 1;

%e 0, 1;

%e 0, 1, 1;

%e 0, 0, 2, 1;

%e 0, 0, 2, 3, 1;

%e 0, 0, 0, 5, 4, 1;

%e 0, 0, 0, 5, 9, 5, 1;

%e 0, 0, 0, 0, 14, 14, 6, 1;

%p 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

%Y Cf. A008313, A009766, A039598, A039599

%K nonn,tabl

%O 0,9

%A _Philippe Deléham_, Aug 17 2006, corrected Sep 15 2006

%E Corrected formula . - _Philippe Deléham_, Oct 16 2008

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Last modified January 19 21:52 EST 2019. Contains 319310 sequences. (Running on oeis4.)