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A007971 INVERTi transform of central trinomial coefficients (A002426). 9

%I

%S 0,1,2,2,4,8,18,42,102,254,646,1670,4376,11596,31022,83670,227268,

%T 621144,1706934,4713558,13072764,36398568,101704038,285095118,

%U 801526446,2259520830,6385455594,18086805002,51339636952,146015545604

%N INVERTi transform of central trinomial coefficients (A002426).

%C 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]

%H G. C. Greubel, <a href="/A007971/b007971.txt">Table of n, a(n) for n = 0..1000</a>

%F A002426(n) = Sum_{i=1..n} a(i)*A002426(n-i), n>0. - _Michael Somos_, Jun 14 2000

%F G.f.: 1 - sqrt(1 - 2*x - 3*x^2). - _Michael Somos_, Jun 14 2000

%F 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

%F 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

%F 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

%F a(n+2) = 2 * A001006(n). - _Michael Somos_, Jun 14 2000

%F For n>1, a(n) = 2 * (A005043(n-1) + A005043(n-2)). - _Ralf Stephan_, Jul 06 2003

%F 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

%F n*a(n) + (-2*n+3)*a(n-1) + *(-n+3)*a(n-2) = 0. - _R. J. Mathar_, Sep 06 2016

%e 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 + ...

%t CoefficientList[Series[1-Sqrt[1-2x-3x^2],{x,0,40}],x] (* _Harvey P. Dale_, Dec 17 2012 *)

%o (PARI) x='x+O('x^50); concat([0], Vec(1 - sqrt(1 - 2*x - 3*x^2))) \\ _G. C. Greubel_, Feb 26 2017

%Y Cf. A001006, A002426, A005043.

%Y Cf. A025227.

%K nonn

%O 0,3

%A David Dumas (dumas(AT)TCNJ.EDU)

%E Name corrected by _Michael Somos_, Mar 23 2012

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Last modified October 22 06:02 EDT 2018. Contains 316432 sequences. (Running on oeis4.)