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A360426 Number of permutations of [2n] having exactly n alternating up/down runs where the first run is not a down run.

%I A360426 #32 Feb 18 2023 03:41:55
%S A360426 1,1,6,118,4788,325446,33264396,4766383420,911323052520,
%T A360426 224136553339270,68929638550210620,25914939202996628148,
%U A360426 11693626371194331008088,6236691723226152102621084,3881046492003600271067466744,2786922888404654795314066258488,2287283298159853722760705106305488
%N A360426 Number of permutations of [2n] having exactly n alternating up/down runs where the first run is not a down run.
%C A360426 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 A360426 Alois P. Heinz, <a href="/A360426/b360426.txt">Table of n, a(n) for n = 0..228</a>
%F A360426 a(n) = A008970(2n,n) = (1/2) * A059427(2n,n) for n>=1.
%F A360426 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 A360426 a(0) = 1: (), the empty permutation.
%e A360426 a(1) = 1: 12.
%e A360426 a(2) = 6: 1243, 1342, 1432, 2341, 2431, 3421.
%e A360426 a(3) = 118: 123546, 123645, 124356, ..., 564123, 564213, 564312.
%p A360426 b:= proc(n, k) option remember; `if`(n<2, 0, `if`(k=1, 1,
%p A360426       k*b(n-1, k) + 2*b(n-1, k-1) + (n-k)*b(n-1, k-2)))
%p A360426     end:
%p A360426 a:= n-> `if`(n=0, 1, b(2*n, n)):
%p A360426 seq(a(n), n=0..17);
%Y A360426 Cf. A000111, A008970, A059427.
%K A360426 nonn
%O A360426 0,3
%A A360426 _Alois P. Heinz_, Feb 08 2023