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A122898
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Expansion of (sqrt(21*x^2 - 10*x + 1) + 7*x - 1) / (2*x*(1 - 7*x)).
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7
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1, 6, 37, 233, 1491, 9660, 63195, 416610, 2763595, 18426026, 123375927, 829053197, 5588050069, 37764371676, 255800207277, 1736181639585, 11804962371795, 80394249836010, 548283258074895, 3744067955618403, 25596986050620681
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OFFSET
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0,2
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COMMENTS
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3rd binomial transform of C(2n+1,n+1) (A001700); 5th binomial transform of C(n,floor(n/2)) (A001405); 7th binomial transform of (-1)^n*A000108(n) = A168491(n). Hankel transform is (1,1,1,.....). Row sums of Riordan array (1/(1+5x+x^2), x/(1+5x+x^2))^(-1). Counts Motzkin paths with 5 colors for horizontal steps. [Corrected by Philippe Deléham, Nov 29 2009]
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LINKS
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FORMULA
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a(n) = Sum{k=0..n, 3^(n-k)C(n,k)C(2k+1,k+1)}.
a(n) = Sum{k=0..n, 5^(n-k)C(n,k)C(k,floor(k/2))}.
a(n) = Sum{k=0..n, 7^(n-k)C(n,k)*(-1)^k*C(k)} where C(n)=A000108(n).
a(n) = Sum{k=0..n, sum{j=0..n, 3^(n-j)*C(n,k)*C(n-k,j-k)*C(j+1,k+1)}}.
G.f.: 1/(1-6x-x^2/(1-5x-x^2/(1-5x-x^2/(1-5x-x^2/(1-...(continued fraction). - Philippe Deléham, Nov 28 2009
D-finite with recurrence: (n+1)*a(n) = 2*(5*n+1)*a(n-1) - 21*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
G.f.: G(0)/(2*x) - 1/(2*x), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1-3*x) - 2*x*(1-3*x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-3*x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 24 2013
a(n) = (-1)^n*(GegenbauerC(n,-n,5/2) - GegenbauerC(n-1,-n,5/2)). - Peter Luschny, May 13 2016
E.g.f.: exp(5*x)*(BesselI(0,2*x) + BesselI(1,2*x)). - Ilya Gutkovskiy, Sep 20 2017
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MAPLE
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a := n -> (-1)^n*simplify(GegenbauerC(n, -n, 5/2) - GegenbauerC(n-1, -n, 5/2)):
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MATHEMATICA
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CoefficientList[Series[(Sqrt[21*x^2-10*x+1]+7*x-1)/(2*x*(1-7*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)
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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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