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A084605
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G.f.: 1/(1-2x-15x^2)^(1/2); also, a(n) is the central coefficient of (1+x+4x^2)^n.
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6
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1, 1, 9, 25, 145, 561, 2841, 12489, 60705, 281185, 1353769, 6418809, 30917041, 148331665, 716698425, 3462260265, 16786700865, 81464917185, 396215601225, 1929237099225, 9408084660945, 45928695279345, 224476389327705
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OFFSET
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0,3
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COMMENTS
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Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U (or D) steps come in four colors. - N-E. Fahssi, Mar 30 2008
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
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FORMULA
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E.g.f.: exp(x)*BesselI(0, 4*x). - Vladeta Jovovic, Aug 20 2003
a(n) is also the central coefficient of (4+x+x^2)^n; a(n)=sum_{k=0..n} 3^(n-k) C(n,k) T(k,n), where T(k,n) is the triangle of trinomial coefficients = Coefficient of x^n of (1+x+x^2)^k : A027907 - N-E. Fahssi, Mar 30 2008
a(n) = (1/Pi)*integral(x=-2..2, (2*x+1)^n/sqrt((2-x)*(2+x))). [Peter Luschny, Sep 12 2011]
a(n+2)=( (2*n+3)*a(n+1) + 15*(n+1)*a(n) )/(n+2); a(0)=a(1)=1 - Sergei N. Gladkovskii, Aug 01 2012
a(n) ~ 5^(n+1/2)/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Oct 14 2012
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MATHEMATICA
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Table[n!*SeriesCoefficient[E^x*BesselI[0, 4*x], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)
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PROG
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(PARI) for(n=0, 30, t=polcoeff((1+x+4*x^2)^n, n, x); print1(t", "))
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CROSSREFS
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Cf. A002426, A084600-A084604, A084606-A084615.
Sequence in context: A139818 A146365 A146373 * A098773 A089998 A014728
Adjacent sequences: A084602 A084603 A084604 * A084606 A084607 A084608
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Jun 01 2003
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STATUS
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approved
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