A276852 - OEIS

%I #29 Oct 11 2016 15:54:08

%S 0,1,2,7,28,121,560,2677,13230,66742,343092,1788681,9439870,50321865,

%T 270594896,1465941763,7993664588,43839212778,241650560756,

%U 1338084935826,7439615051328,41516113036777,232452845782308,1305500166481715,7352433083806020,41514430735834714

%N Number of positive walks with n steps {-3,-2,-1,1,2,3} starting at the origin, ending at altitude 1, and staying strictly above the x-axis.

%H Alois P. Heinz, <a href="/A276852/b276852.txt">Table of n, a(n) for n = 0..1292</a>

%H C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, <a href="https://arxiv.org/abs/1609.06473">Explicit formulas for enumeration of lattice paths: basketball and the kernel method</a>, arXiv preprint arXiv:1609.06473 [math.CO], 2016.

%t walks[n_, k_, h_] = 0;

%t walks[1, k_, h_] := Boole[0 < k <= h];

%t walks[n_, k_, h_] /; n >= 2 && k > 0 := walks[n, k, h] = Sum[walks[n - 1, k - x, h], {x, h}] + Sum[walks[n - 1, k + x, h], {x, h}];

%t (* walks represents the number of positive walks with n steps {-h, -h+1, ... -1, 1, ..., h} that end at altitude k *)

%t A276852[n_] := (Do[walks[m, k, 3], {m, n}, {k, 3 m}]; walks[n, 1, 3]) (* _Davin Park_, Oct 10 2016 *)

%Y Cf. A276901, A276902, A276903, A276904.

%K nonn,walk

%O 0,3

%A _Michael Wallner_, Sep 21 2016