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%I #23 Jan 01 2024 02:33:30
%S 1,1,3,12,57,301,1707,10191,63244,404503,2650293,17709684,120288313,
%T 828352036,5771747783,40625485570,288482116987,2064429518054,
%U 14874855533504,107832596542894,785986247826371,5757192302807027,42357833323697589,312901369167191854,2319946973815289676
%N Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of 2 n steps taken from {(-1, 1), (1, -1), (1, 0)}.
%C From _Henri Mühle_, Nov 27 2020: (Start)
%C a(n) is also the sum over all parabolic Catalan objects associated with parabolic quotients of the symmetric group S_n. The parabolic quotients of S_n are indexed by compositions of n. If alpha=(a_1,a_2,..., a_r) is a composition of n, consider the Dyck path v_alpha = N^{a_1}E^{a_1}N^{a_2}E^{a_2}...N^{a_r}E^{a_r}. The number of parabolic Catalan objects Cat(alpha) associated with alpha equals the number of Dyck paths of semilength n weakly above v_alpha.
%C For instance, if n=3, there are four compositions: alpha_1=(3), alpha_2=(2,1), alpha_3=(1,2), alpha_4=(1,1,1). Then, a(3) = Sum_{i=1..4} Cat(alpha_i) = 1+3+3+5 = 12.
%C (End)
%H Nantel Bergeron, Cesar Ceballos, Vincent Pilaud, <a href="https://arxiv.org/abs/1807.03044">Hopf dreams</a>, arXiv:1807.03044 [math.CO], 2018. See p. 19.
%H M. Bousquet-Mélou and M. Mishna, <a href="https://arxiv.org/abs/0810.4387">Walks with small steps in the quarter plane</a>, arXiv:0810.4387 [math.CO], 2008-2009.
%H Cesar Ceballos, Wenjie Fang, Henri Mühle, <a href="https://arxiv.org/abs/1903.08515">The Steep-Bounce zeta map in Parabolic Cataland</a>, arXiv:1903.08515 [math.CO], 2019. See pp. 32ff.
%t 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, j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n]]; Table[Sum[aux[0, k, 2 n], {k, 0, 2 n}], {n, 0, 25}]
%K nonn,walk
%O 0,3
%A _Manuel Kauers_, Nov 18 2008
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