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A007971 INVERTi transform of central trinomial coefficients (A002426). 10
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: A175195 A139800 A168058 * A126068 A167022 A168055

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 January 22 01:47 EST 2019. Contains 319351 sequences. (Running on oeis4.)