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A360426
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Number of permutations of [2n] having exactly n alternating up/down runs where the first run is not a down run.
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3
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1, 1, 6, 118, 4788, 325446, 33264396, 4766383420, 911323052520, 224136553339270, 68929638550210620, 25914939202996628148, 11693626371194331008088, 6236691723226152102621084, 3881046492003600271067466744, 2786922888404654795314066258488, 2287283298159853722760705106305488
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OFFSET
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0,3
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COMMENTS
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Number of permutations of [2n] such that the differences have n runs with the same signs where the first run does not have negative signs.
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LINKS
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FORMULA
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a(n) ~ c * d^n * n!^2 / n, where d = 3.421054620671187024940215794079585351303138828348... (same as for A291677 and A303159) and c = 0.23613698601500409294656476488227001191406... - Vaclav Kotesovec, Feb 18 2023
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EXAMPLE
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a(0) = 1: (), the empty permutation.
a(1) = 1: 12.
a(2) = 6: 1243, 1342, 1432, 2341, 2431, 3421.
a(3) = 118: 123546, 123645, 124356, ..., 564123, 564213, 564312.
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MAPLE
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b:= proc(n, k) option remember; `if`(n<2, 0, `if`(k=1, 1,
k*b(n-1, k) + 2*b(n-1, k-1) + (n-k)*b(n-1, k-2)))
end:
a:= n-> `if`(n=0, 1, b(2*n, n)):
seq(a(n), n=0..17);
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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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