cp's OEIS Frontend

A151282 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, -1), (-1, 0), (0, 1), (1, 1)}.

%I A151282 #38 Aug 06 2024 04:49:51
%S A151282 1,2,6,18,58,190,638,2170,7474,25974,90982,320738,1137002,4049838,
%T A151282 14485326,52001290,187292514,676546790,2450311862,8895769714,
%U A151282 32366225562,117995832990,430960312862,1576675041434,5777325893266,21200338220630,77901645076998,286615385651970,1055762834791114,3893279267979662
%N A151282 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, -1), (-1, 0), (0, 1), (1, 1)}.
%C A151282 From _Paul Barry_, Jan 19 2009: (Start)
%C A151282 Hankel transform is 2^C(n+1,2).
%C A151282 Row sums of Riordan array ((1-2x)/(1-x+2x^2),x(1-x)/(1-x+2x^2))^{-1}.
%C A151282 G.f.: 1/(1-2x-2x^2/(1-x-2x^2/(1-x-2x^2/(1-x-2x^2/(1-...))))) (continued fraction).
%C A151282 First column of Riordan array ((1-x)/(1+x+2x^2),x/(1+x+2x^2))^{-1}. (End)
%H A151282 Robert Israel, <a href="/A151282/b151282.txt">Table of n, a(n) for n = 0..200</a>
%H A151282 A. Bostan, <a href="https://citeseerx.ist.psu.edu/pdf/749aef4c6f3668e652b5074e5268346ccecc88c9">Computer Algebra for Lattice Path Combinatorics</a>, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.
%H A151282 A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, <a href="http://arxiv.org/abs/0811.2899">ArXiv 0811.2899</a>.
%H A151282 Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, <a href="https://arxiv.org/abs/2001.00393">Stieltjes moment sequences for pattern-avoiding permutations</a>, arXiv:2001.00393 [math.CO], 2020.
%H A151282 M. Bousquet-Mélou and M. Mishna, 2008. Walks with small steps in the quarter plane, <a href="http://arxiv.org/abs/0810.4387">ArXiv 0810.4387</a>.
%F A151282 Conjecture: (n+1)*a(n)-3*(2n+1)*a(n-1) +(n+10)*a(n-2) +28(n-2)*a(n-3)=0. - _R. J. Mathar_, Dec 08 2011
%F A151282 a(n) ~ sqrt(1348+953*sqrt(2)) * (1+2*sqrt(2))^n/(2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Aug 14 2013
%e A151282 G.f. = 1 + 2*x + 6*x^2 + 18*x^3 + 58*x^4 + 190*x^5 + 638*x^6 + 2170*x^7 + ...
%p A151282 b:= proc(n, l) option remember; `if`(-1 in {l[]}, 0, `if`(n=0, 1,
%p A151282       add(b(n-1, l+d), d=[[-1, -1], [-1, 0], [0, 1], [1, 1]])))
%p A151282     end:
%p A151282 a:= n-> b(n, [0$2]):
%p A151282 seq(a(n), n=0..40);  # _Alois P. Heinz_, Feb 18 2013
%t A151282 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[i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[Sum[aux[i, j, n], {i, 0, n}, {j, 0, n}], {n, 0, 25}]
%K A151282 nonn,walk
%O A151282 0,2
%A A151282 _Manuel Kauers_, Nov 18 2008