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A120730
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Another version of Catalan triangle A009766.
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24
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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
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OFFSET
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0,9
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COMMENTS
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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 DELEHAM, 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
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LINKS
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Table of n, a(n) for n=0..65.
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FORMULA
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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 DELEHAM, Oct 18 2008]
Sum_{k, 0<=k<=n}T(n,k)^3 = A003161(n). [From Philippe DELEHAM, Oct 18 2008]
Sum_{k, 0<=k<=n}T(n,k)^4 = A129123(n). [From Philippe DELEHAM, 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 DELEHAM, Feb 10 2009]
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EXAMPLE
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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;
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MAPLE
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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
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CROSSREFS
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Cf. A008313, A009766, A039598, A039599
Sequence in context: A055791 A191400 A168315 * A122851 A064301 A199881
Adjacent sequences: A120727 A120728 A120729 * A120731 A120732 A120733
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KEYWORD
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nonn,tabl
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AUTHOR
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Philippe DELEHAM, Aug 17 2006, corrected Sep 15 2006
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EXTENSIONS
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Corrected formula . - Philippe DELEHAM, Oct 16 2008
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STATUS
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approved
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