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A007971 INVERTi transform of central trinomial coefficients (A002426). 13
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
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. A025227.
Sequence in context: A369289 A139800 A168058 * A126068 A167022 A168055
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 March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)