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A084601
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Coefficients of 1/(1-2x-7x^2)^(1/2); also, a(n) is the central coefficient of (1+x+2x^2)^n.
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11
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1, 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693, 243954113, 906758785, 3375949829, 12587460557, 46995614449, 175669746209, 657370655045, 2462383495357, 9232029156001
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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), U (or D) can have 2 colors. - N-E. Fahssi, Feb 05 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)2^k}. - Paul Barry, Aug 26 2004
Sum[k=0..n, Trinomial(k, n) Binomial(n, k) ], with Trinomial=A027907. - Ralf Stephan, Jan 28 2005
a(n) is also the central coefficient of (2+x+x^2)^n; a(n)=sum_{k=0..n} 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, Feb 05 2008
a(n+2)=( (2*n+3)*a(n+1) + 7*(n+1)*a(n) )/(n+2); a(0)=a(1)=1 - Sergei N. Gladkovskii, Aug 01 2012
G.f.: G(0), where G(k)= 1 + x*(2+7*x)*(4*k+1)/( 4*k+2 - x*(2+7*x)*(4*k+2)*(4*k+3)/(x*(2+7*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2013
a(n) ~ sqrt(8+2*sqrt(2)) * (1+2*sqrt(2))^n / (4*sqrt(Pi*n)). - Vaclav Kotesovec, May 09 2014
a(n) = hypergeom([1/2 - n/2, -n/2], [1], 8). - Peter Luschny, Mar 18 2018
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MAPLE
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a := n -> hypergeom([1/2 - n/2, -n/2], [1], 8):
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MATHEMATICA
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CoefficientList[Series[1/Sqrt[1-2x-7x^2], {x, 0, 30}], x] (* Harvey P. Dale, Sep 18 2011 *)
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PROG
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(PARI) for(n=0, 30, t=polcoeff((1+x+2*x^2)^n, n, x); print1(t", "))
(Maxima) a(n):=coeff(expand((1+x+2*x^2)^n), x, n);
makelist(a(n), n, 0, 12); /* Emanuele Munarini, Mar 02 2011 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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