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A333017
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Twice the total area of all (open or closed) Deutsch paths of length n.
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2
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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MAPLE
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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);
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MATHEMATICA
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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}]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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