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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.
3

%I #32 Feb 18 2023 03:41:55

%S 1,1,6,118,4788,325446,33264396,4766383420,911323052520,

%T 224136553339270,68929638550210620,25914939202996628148,

%U 11693626371194331008088,6236691723226152102621084,3881046492003600271067466744,2786922888404654795314066258488,2287283298159853722760705106305488

%N Number of permutations of [2n] having exactly n alternating up/down runs where the first run is not a down run.

%C 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.

%H Alois P. Heinz, <a href="/A360426/b360426.txt">Table of n, a(n) for n = 0..228</a>

%F a(n) = A008970(2n,n) = (1/2) * A059427(2n,n) for n>=1.

%F 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

%e a(0) = 1: (), the empty permutation.

%e a(1) = 1: 12.

%e a(2) = 6: 1243, 1342, 1432, 2341, 2431, 3421.

%e a(3) = 118: 123546, 123645, 124356, ..., 564123, 564213, 564312.

%p b:= proc(n, k) option remember; `if`(n<2, 0, `if`(k=1, 1,

%p k*b(n-1, k) + 2*b(n-1, k-1) + (n-k)*b(n-1, k-2)))

%p end:

%p a:= n-> `if`(n=0, 1, b(2*n, n)):

%p seq(a(n), n=0..17);

%Y Cf. A000111, A008970, A059427.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Feb 08 2023