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 A129147 Expansion of c(x(1+2x)), c(x) the g.f. of A000108. 1
 1, 1, 4, 13, 52, 214, 928, 4141, 18940, 88258, 417616, 2001058, 9690184, 47348812, 233158144, 1155900541, 5764510060, 28898899594, 145556001136, 736206912982, 3737768204344, 19042072755124, 97313398530496, 498737257238482, 2562773039735896, 13200732624526804, 68148459129343648 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Hankel transform of a(n) is A047656(n+1)=3^C(n+1,2). In general, the Hankel transform of the expansion of c(x(1+r*x)) is (r+1)^C(n+1,2). Number of paths weakly above X-axis from (0,0) to (0,2n) using steps (1,1), (1,-1) and two colors of (3,1). - David Scambler, Jun 21 2013 REFERENCES Barry, Paul; Hennessy, Aoife Four-term recurrences, orthogonal polynomials and Riordan arrays. J. Integer Seq. 15 (2012), no. 4, Article 12.4.2, 19 pp. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 FORMULA a(n)=sum{k=0..n, C(k,n-k)*2^(n-k)*C(k)}; a(n)=(1/(2*pi))*int(x^n*sqrt(8+4x-x^2)/(x+2),x,2-2*sqrt(3),2+2*sqrt(3)); Conjecture: (n+1)*a(n) +2*(2-n)*a(n-1) +4*(5-4n)*a(n-2) +16*(2-n)*a(n-3)=0. - R. J. Mathar, Dec 14 2011 G.f.: Q(0), where Q(k)= 1 + (4*k+1)*x*(1+2*x)/(k + 1 - x*(1+2*x)*(2*k+2)*(4*k+3)/(2*x*(1+2*x)*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013 a(n) ~ sqrt(3-sqrt(3)) * (2*(1+sqrt(3)))^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 01 2014 MATHEMATICA CoefficientList[Series[(1-Sqrt[1-4*x*(1+2*x)])/(2*x*(1+2*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *) PROG (PARI) x='x+O('x^66); C(x)=(1-sqrt(1-4*x))/(2*x); Vec(C(x*(1+2*x))) \\ Joerg Arndt, May 15 2013 CROSSREFS Sequence in context: A323791 A149464 A307398 * A151476 A140454 A149465 Adjacent sequences:  A129144 A129145 A129146 * A129148 A129149 A129150 KEYWORD easy,nonn AUTHOR Paul Barry, Apr 01 2007 STATUS approved

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Last modified May 24 16:36 EDT 2022. Contains 354042 sequences. (Running on oeis4.)