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A014531
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Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 2nd column from the center.
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14
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1, 3, 10, 30, 90, 266, 784, 2304, 6765, 19855, 58278, 171106, 502593, 1477035, 4343160, 12778152, 37616427, 110797569, 326527350, 962803170, 2840372304, 8383467708, 24755608584, 73133433800, 216143407675, 639062383401
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OFFSET
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1,2
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COMMENTS
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Number of "up" steps in all Motzkin paths of length n+1. E.g. a(2)=3 because in the four Motzkin paths of length 3, HHH, HUD, UDH and UHD, where H=(1,0), U=(1,1), D=(1,-1), we have altogether three U steps. - Emeric Deutsch, Dec 26 2003
a(n) = number of paths in the half-plane x>=0, from (0,0) to (n+1,2), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=2, we have the 3 paths: UUH, HUU, UHU. - José Luis Ramírez Ramírez, Apr 19 2015
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
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LINKS
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FORMULA
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E.g.f.: exp(x)*(2*x*BesselI(1, 2*x)+(x-2)*BesselI(2, 2*x))/x. - Vladeta Jovovic, Aug 21 2003
G.f.: [1-2z-z^2-(1-z)q]/(2z^3q), where q=sqrt(1-2z-3z^2). - Emeric Deutsch, Dec 26 2003
a(n) = sum{k=0..n+1, C(n+1,k)*C(n-k+1,k+2)} - Paul Barry, Sep 20 2004
D-finite with recurrence (n+3)*(n-1)*a(n) -(n+1)*(2n+1)*a(n-2)-3*n*(n+1)*a(n-2)=0. - R. J. Mathar, Dec 08 2011
a(n) = n*(n+1)*hypergeom([(1-n)/2, 1-n/2], [3], 4)/2. - Peter Luschny, Nov 23 2014
a(n) = GegenbauerC(n-1, -n-1, -1/2). - Peter Luschny, May 09 2016
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MAPLE
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seq( add(binomial(i+1, k)*binomial(i-k+1, k+2), k=0..floor(i/2)), i=1..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
a := n -> simplify(GegenbauerC(n-1, -n-1, -1/2)):
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MATHEMATICA
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Table[Sum[Binomial[i + 1, k]*Binomial[i - k + 1, k + 2], {k, 0, Floor[i/2]}], {i, 30}] (* Michael De Vlieger, Apr 20 2015 *)
Table[GegenbauerC[n - 1, -n - 1, -1/2], {n, 1, 50}] (* G. C. Greubel, Feb 28 2017 *)
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PROG
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(Sage)
a = lambda n: n*(n+1)*hypergeometric([(1-n)/2, 1-n/2], [3], 4)/2
(PARI) for(n=1, 25, print1(sum(k=0, n+1, binomial(n+1, k)*binomial(n-k+1, k+2)), ", ")) \\ G. C. Greubel, Feb 28 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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