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A098329
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Expansion of 1/(1-2x-31x^2)^(1/2).
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2
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1, 1, 17, 49, 481, 2081, 16241, 85457, 600769, 3489601, 23391697, 143000177, 938797729, 5897385313, 38397492017, 244866166289, 1590355308929, 10231490804353, 66456634775441, 429898281869489, 2795449543782241, 18150017431150241, 118194927388259057, 769438418283309649
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OFFSET
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0,3
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COMMENTS
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Central coefficient of (1+x+8x^2)^n. 7th binomial transform of 2^n*LegendreP(n,-3) (signed version of A084773).
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps can have 8 colors. - N-E. Fahssi, Mar 31 2008
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LINKS
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FORMULA
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a(n) = sum{k=0..floor(n/2), binomial(n-k, k)*binomial(n, k)*8^k}.
E.g.f.: exp(x)*BesselI(0, 4*sqrt(2)*x)
Recurrence: n*a(n) = (2*n-1)*a(n-1) + 31*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ sqrt(8+sqrt(2))*(1+4*sqrt(2))^n/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012
a(n) = hypergeom([1/2 - n/2, -n/2], [1], 32). - Peter Luschny, Mar 18 2018
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MATHEMATICA
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Table[SeriesCoefficient[1/Sqrt[1-2*x-31*x^2], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)
CoefficientList[Series[1/Sqrt[1-2x-31x^2], {x, 0, 30}], x] (* Harvey P. Dale, May 14 2017 *)
a[n_] := Hypergeometric2F1[1/2 - n/2, -n/2, 1, 32];
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PROG
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(PARI) x='x+O('x^66); Vec(1/(1-2*x-31*x^2)^(1/2)) \\ Joerg Arndt, May 11 2013
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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