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 A007971 INVERTi transform of central trinomial coefficients (A002426). 9
 0, 1, 2, 2, 4, 8, 18, 42, 102, 254, 646, 1670, 4376, 11596, 31022, 83670, 227268, 621144, 1706934, 4713558, 13072764, 36398568, 101704038, 285095118, 801526446, 2259520830, 6385455594, 18086805002, 51339636952, 146015545604 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Number of paths of a walk on the integers, allowing steps of size 0, +1, and -1, which return to the starting point for the first time at time n. [David P. Sanders (dps(AT)fciencias.unam.mx), May 04 2009] LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 FORMULA A002426(n) = Sum_{i=1..n} a(i)*A002426(n-i), n>0. - Michael Somos, Jun 14 2000 G.f.: 1 - sqrt(1 - 2*x - 3*x^2). - Michael Somos, Jun 14 2000 a(0)=0, a(1)=1, a(2)=2, then a(n)= (1/2) *(a(1)*a(n-1)+a(2)*a(n-2)+....+a(n-1)*a(1)). - Benoit Cloitre, Oct 24 2003 a(n) = 2^(1-n)*Sum_{k=1..n} (binomial(k,n-k)*a000108(k-1)*3^(n-k)), n>0. - Vladimir Kruchinin, Feb 05 2011 G.f.: 1-sqrt(1-2*x-3*(x^2)) =  x/G(0) ; G(k) = 1-2*x/(1+x/(1+x/(1-2*x/(1-x/(2-x/G(k+1)))))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 11 2011 a(n+2) = 2 * A001006(n). - Michael Somos, Jun 14 2000 For n>1, a(n) = 2 * (A005043(n-1) + A005043(n-2)). - Ralf Stephan, Jul 06 2003 0 = a(n) * (9*a(n+1) + 15*a(n+2) - 12*a(n+3)) + a(n+1) * (-3*a(n+1) + 10*a(n+2) - 5*a(n+3)) + a(n+2) * (a(n+2) + a(n+3)) for all n>0. - Michael Somos, Jan 25 2014 n*a(n) + (-2*n+3)*a(n-1) + *(-n+3)*a(n-2) = 0. - R. J. Mathar, Sep 06 2016 EXAMPLE G.f. = x + 2*x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 18*x^6 + 42*x^7 + 102*x^8 + 254*x^9 + ... MATHEMATICA CoefficientList[Series[1-Sqrt[1-2x-3x^2], {x, 0, 40}], x] (* Harvey P. Dale, Dec 17 2012 *) PROG (PARI) x='x+O('x^50); concat([0], Vec(1 - sqrt(1 - 2*x - 3*x^2))) \\ G. C. Greubel, Feb 26 2017 CROSSREFS Cf. A001006, A002426, A005043. Cf. A025227. Sequence in context: * A126068 A167022 A168055 A005702 A095335 A283117 Adjacent sequences:  A007968 A007969 A007970 * A007972 A007973 A007974 KEYWORD nonn AUTHOR David Dumas (dumas(AT)TCNJ.EDU) EXTENSIONS Name corrected by Michael Somos, Mar 23 2012 STATUS approved

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Last modified August 15 18:49 EDT 2018. Contains 313779 sequences. (Running on oeis4.)