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 A151498 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)} 0
 1, 1, 3, 12, 57, 301, 1707, 10191, 63244, 404503, 2650293, 17709684, 120288313, 828352036, 5771747783, 40625485570, 288482116987, 2064429518054, 14874855533504, 107832596542894, 785986247826371, 5757192302807027, 42357833323697589, 312901369167191854, 2319946973815289676 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS From Henri Mühle, Nov 27 2020: (Start) 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. 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. (End) LINKS Table of n, a(n) for n=0..24. Nantel Bergeron, Cesar Ceballos, Vincent Pilaud, Hopf dreams, arXiv:1807.03044 [math.CO], 2018. See p. 19. M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009. Cesar Ceballos, Wenjie Fang, Henri Mühle, The Steep-Bounce zeta map in Parabolic Cataland, arXiv:1903.08515 [math.CO], 2019. See pp. 32ff. 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, 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}] CROSSREFS Sequence in context: A166991 A276366 A243521 * A103370 A094149 A291695 Adjacent sequences: A151495 A151496 A151497 * A151499 A151500 A151501 KEYWORD nonn,walk AUTHOR Manuel Kauers, Nov 18 2008 STATUS approved

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Last modified September 29 19:17 EDT 2023. Contains 365776 sequences. (Running on oeis4.)