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 A333017 Twice the total area of all (open or closed) Deutsch paths of length n. 2
 0, 1, 6, 25, 90, 306, 1004, 3226, 10218, 32043, 99748, 308787, 951772, 2923563, 8955342, 27368895, 83484042, 254244033, 773219196, 2348780937, 7127522136, 21609615822, 65465845254, 198189732798, 599624708588, 1813169256151, 5480019176754, 16555101318735 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Deutsch paths (named after their inventor Emeric Deutsch by Helmut Prodinger) are like Dyck paths where down steps can get to all lower levels. Open paths can end at any level, whereas closed paths have to return to the lowest level zero at the end. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..2094 Helmut Prodinger, Deutsch paths and their enumeration, arXiv:2003.01918 [math.CO], 2020. See p. 8. Wikipedia, Counting lattice paths MAPLE b:= proc(x, y) option remember; `if`(x=0, [1, 0], add((p-> p+[0, (2*y-j)*p[1]])(b(x-1, y-j)), j=[\$1..y, -1])) end: a:= n-> b(n, 0)[2]: seq(a(n), n=0..30); # second Maple program: a:= proc(n) option remember; `if`(n<4, [0, 1, 6, 25][n+1], ((1045*n^2-4419*n-9646)*a(n-1)-3*(1133*n^2-4679*n-1756)* a(n-2)+9*(127*n^2-475*n+480)*a(n-3)+27*(210*n-439)* (n-3)*a(n-4))/((n+3)*(83*n-677))) end: seq(a(n), n=0..30); MATHEMATICA a = DifferenceRoot[Function[{y, n}, {(-10827 - 16497 n - 5670 n^2) y[n] + (-5508 - 4869 n - 1143 n^2) y[n+1] + (-7032 + 13155 n + 3399 n^2) y[n+2] + (10602 - 3941 n - 1045 n^2) y[n+3] + (7 + n)(-345 + 83 n) y[n+4] == 0, y[0] == 0, y[1] == 1, y[2] == 6, y[3] == 25}]]; a /@ Range[0, 30] (* Jean-François Alcover, Mar 12 2020 *) CROSSREFS Cf. A001006, A005043, A330169. Sequence in context: A000392 A365531 A099948 * A277973 A143815 A209241 Adjacent sequences: A333014 A333015 A333016 * A333018 A333019 A333020 KEYWORD nonn AUTHOR Alois P. Heinz, Mar 05 2020 STATUS approved

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Last modified August 6 17:27 EDT 2024. Contains 374981 sequences. (Running on oeis4.)