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A007971 INVERTi transform of central trinomial coefficients (A002426).

%I A007971 #54 Feb 20 2025 03:21:54
%S A007971 0,1,2,2,4,8,18,42,102,254,646,1670,4376,11596,31022,83670,227268,
%T A007971 621144,1706934,4713558,13072764,36398568,101704038,285095118,
%U A007971 801526446,2259520830,6385455594,18086805002,51339636952,146015545604
%N A007971 INVERTi transform of central trinomial coefficients (A002426).
%C A007971 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 A007971 G. C. Greubel, <a href="/A007971/b007971.txt">Table of n, a(n) for n = 0..1000</a>
%H A007971 Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2502.13673">Pseudo-involutions in the Riordan group and Chebyshev polynomials</a>, arXiv:2502.13673 [math.CO], 2025.
%F A007971 A002426(n) = Sum_{i=1..n} a(i)*A002426(n-i), n>0. - _Michael Somos_, Jun 14 2000
%F A007971 G.f.: 1 - sqrt(1 - 2*x - 3*x^2). - _Michael Somos_, Jun 14 2000
%F A007971 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 A007971 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 A007971 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 A007971 a(n+2) = 2 * A001006(n). - _Michael Somos_, Jun 14 2000
%F A007971 For n>1, a(n) = 2 * (A005043(n-1) + A005043(n-2)). - _Ralf Stephan_, Jul 06 2003
%F A007971 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 A007971 n*a(n) + (-2*n+3)*a(n-1) + *(-n+3)*a(n-2) = 0. - _R. J. Mathar_, Sep 06 2016
%e A007971 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 A007971 CoefficientList[Series[1-Sqrt[1-2x-3x^2],{x,0,40}],x] (* _Harvey P. Dale_, Dec 17 2012 *)
%t A007971 a[1]:=1;a[2]:=2;a[n_]:=a[n]=1/2 Sum[a[k] a[n-k],{k,1,n-1}];
%t A007971 Join[{0},Map[a,Range[24]]] (* _Oliver Seipel_, Nov 03 2024, after Schröder 1870 *)
%o A007971 (PARI) x='x+O('x^50); concat([0], Vec(1 - sqrt(1 - 2*x - 3*x^2))) \\ _G. C. Greubel_, Feb 26 2017
%Y A007971 Cf. A001006, A002426, A005043.
%Y A007971 Cf. A025227.
%K A007971 nonn
%O A007971 0,3
%A A007971 David Dumas (dumas(AT)TCNJ.EDU)
%E A007971 Name corrected by _Michael Somos_, Mar 23 2012