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A014533
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Form array in which n-th row is obtained by expanding (1 + x + x^2)^n and taking the 4th column from the center.
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5
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1, 5, 21, 77, 266, 882, 2850, 9042, 28314, 87802, 270270, 827190, 2520336, 7651632, 23162976, 69954048, 210859245, 634569201, 1907165337, 5725520801, 17172595110, 51465297950, 154135675070, 461366154990, 1380317174145
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OFFSET
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1,2
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COMMENTS
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First differences seem to be in A025182.
a(n-4) = number of paths in the half-plane x >= 0, from (0,0) to (n,4), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=5, we have the 5 paths HUUUU, UHUUU, UUHUU, UUUHU, UUUUH. - 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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Conjecture: -(n+7)*(n-1)*a(n) + (n+3)*(2*n+5)*a(n-1) + 3*(n+3)*(n+2)*a(n-2) = 0. - R. J. Mathar, Feb 25 2015
a(n) = C(6+2*n, n-1)*hypergeom([-n+1, -n-7], [-5/2-n], 1/4).
a(n) = GegenbauerC(n-1, -n-3, -1/2). (End)
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MAPLE
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a := n -> simplify(GegenbauerC(n-1, -n-3, -1/2)):
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MATHEMATICA
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Rest[CoefficientList[Series[x*((1-x-Sqrt[1-2*x-3*x^2])/(2*x^2))^4/(1-x-2*x^2*(1-x-Sqrt[1-2*x-3*x^2])/(2*x^2)), {x, 0, 20}], x]] (* Vaclav Kotesovec, Apr 20 2015 *)
Table[GegenbauerC[n-1, -n - 3, -1/2], {n, 0, 50}] (* G. C. Greubel, Feb 28 2017 *)
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PROG
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(PARI) x='x + O('x^50); Vec(x*((1-x-sqrt(1-2*x-3*x^2))/(2*x^2))^4/(1-x-2*x^2*(1-x-sqrt(1-2*x-3*x^2))/(2*x^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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