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A151281 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, 0), (1, 0), (1, 1)}. 7
1, 2, 6, 16, 48, 136, 408, 1184, 3552, 10432, 31296, 92544, 277632, 824448, 2473344, 7365120, 22095360, 65920000, 197760000, 590790656, 1772371968, 5299916800, 15899750400, 47578857472, 142736572416, 427357700096, 1282073100288, 3840133464064, 11520400392192, 34517383151616, 103552149454848 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From Paul Barry, Jan 26 2009: (Start)

Image of 2^n under A155761. Binomial transform is A129637. Hankel transform is 2^C(n+1,2).

In general, the g.f. of the reversion of x(1+cx)/(1+ax+bx^2) is given by the continued fraction

x/(1-(a-c)x-(b-ac+c^2)x^2/(1-(a-2c)x-(b-ac+c^2)x^2/(1-(a-2c)x-(b-ac+c^2)x^2/(1-.... (End)

LINKS

Robert Israel, Table of n, a(n) for n = 0..2000

P. Barry, A Note on a One-Parameter Family of Catalan-Like Numbers, JIS 12 (2009) 09.5.4

A. Bostan, Computer Algebra for Lattice Path Combinatorics, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.

M. Bousquet-Mélou and M. Mishna, 2008. Walks with small steps in the quarter plane, ArXiv 0810.4387.

A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, ArXiv 0811.2899.

FORMULA

From Paul Barry, Jan 26 2009: (Start)

G.f.: 1/(1-2x-2x^2/(1-2x^2/(1-2x^2/(1-2x^2/(1-2x^2/(1-.... (continued fraction);

G.f.: c(2x^2)/(1-2xc(2x^2))=(sqrt(1-8x^2)+4x-1)/(4x(1-3x));

a(n) = sum{k=0..n, ((k+1)/(n+k+1))*C(n,(n-k)/2)*(1+(-1)^(n-k))*2^((n-k)/ 2)*2^k};

Reversion of x(1+2x)/(1+4x+6x^2). (End)

a(n) = sum{k=0..n, A120730(n,k)*2^k}. - Philippe Deléham, Feb 01 2009

a(2n+2)=3*a(2n+1), a(2n+1)=3*a(2n)-2^n*A000108(n) = 3*a(2n)-A151374(n). - Philippe Deléham, Feb 02 2009

Conjecture: (n+1)*a(n) +3*(-n-1)*a(n-1) +8*(-n+2)*a(n-2) +24*(n-2)* a(n-3)=0. - R. J. Mathar, Nov 26 2012

a(n) ~ 3^n/2. - Vaclav Kotesovec, Feb 13 2014

MAPLE

N:= 1000: # to get terms up to a(N)

S:= series((sqrt(1-8*x^2)+4*x-1)/(4*x*(1-3*x)), x, N+1):

seq(coeff(S, x, j), j=0..N); # Robert Israel, Feb 18 2013

MATHEMATICA

aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, -1 + j, -1 + n] + aux[-1 + i, j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[Sum[aux[i, j, n], {i, 0, n}, {j, 0, n}], {n, 0, 25}]

CROSSREFS

Sequence in context: A148443 A148444 A064190 * A045694 A225178 A129772

Adjacent sequences:  A151278 A151279 A151280 * A151282 A151283 A151284

KEYWORD

nonn,walk

AUTHOR

Manuel Kauers, Nov 18 2008

STATUS

approved

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Last modified October 15 05:56 EDT 2018. Contains 316202 sequences. (Running on oeis4.)