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 A122898 Expansion of (sqrt(21*x^2 - 10*x + 1) + 7*x - 1) / (2*x*(1 - 7*x)). 6
 1, 6, 37, 233, 1491, 9660, 63195, 416610, 2763595, 18426026, 123375927, 829053197, 5588050069, 37764371676, 255800207277, 1736181639585, 11804962371795, 80394249836010, 548283258074895, 3744067955618403, 25596986050620681 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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] Binomial transform of A005573. 7th binomial transform of A168491. - Philippe Deléham, Nov 28 2009 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 FORMULA 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 Recurrence: (n+1)*a(n) = 2*(5*n+1)*a(n-1) - 21*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 19 2012 a(n) ~ 7^(n+1/2)/sqrt(Pi*n). - 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 MAPLE a := n -> (-1)^n*simplify(GegenbauerC(n, -n, 5/2) - GegenbauerC(n-1, -n, 5/2)): seq(a(n), n=0..21); # Peter Luschny, May 13 2016 MATHEMATICA 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 *) CROSSREFS Sequence in context: A076026 A161734 A081570 * A218186 A317629 A081912 Adjacent sequences:  A122895 A122896 A122897 * A122899 A122900 A122901 KEYWORD easy,nonn AUTHOR Paul Barry, Sep 18 2006 STATUS approved

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Last modified January 17 10:30 EST 2019. Contains 319218 sequences. (Running on oeis4.)