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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 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: A300574 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 14 05:17 EST 2018. Contains 318090 sequences. (Running on oeis4.)