A007971 INVERTi transform of central trinomial coefficients (A002426).

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

Offset: 0

Author

David Dumas (dumas(AT)TCNJ.EDU)

Keywords

• nonn

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]

Examples

			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 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.

Crossrefs

Cf. A001006, A002426, A005043.

Cf. A025227.

Programs

Mathematica

CoefficientList[Series[1-Sqrt[1-2x-3x^2],{x,0,40}],x] (* Harvey P. Dale, Dec 17 2012 *)
a[1]:=1;a[2]:=2;a[n_]:=a[n]=1/2 Sum[a[k] a[n-k],{k,1,n-1}];
Join[{0},Map[a,Range[24]]] (* Oliver Seipel, Nov 03 2024, after Schröder 1870 *)

PARI

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

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

Extensions

Name corrected by Michael Somos, Mar 23 2012