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A308962 Number of permutations of [4n] with exactly 2n increasing runs of odd length.

%I A308962 #19 Jul 09 2019 17:33:24
%S A308962 1,17,13930,77296296,1568558071080,84938094880524600,
%T A308962 10128482222614148352960,2336936362896740255803152000,
%U A308962 950622895076910219544822877635200,635598214592375283010356491822548022400,661314598267382330509313757278639302452192000
%N A308962 Number of permutations of [4n] with exactly 2n increasing runs of odd length.
%H A308962 Vaclav Kotesovec, <a href="/A308962/b308962.txt">Table of n, a(n) for n = 0..110</a> (terms 0..69 from Alois P. Heinz)
%F A308962 a(n) = (4n)! * [x^(4n) t^(2n)] t^2/(1-t*x-(1-t^2)*exp(-t*x)). [corrected by _Vaclav Kotesovec_, Jul 09 2019]
%F A308962 a(n) = A097591(4n,2n).
%F A308962 From _Vaclav Kotesovec_, Jul 09 2019: (Start)
%F A308962 a(n)/(4*n)! ~ c * d^n / sqrt(n), where
%F A308962 d = 0.49313160144517183347479521733129940030484540928084707469774969650583707...
%F A308962 c = 3.44699229707824751737600849250650265725079793249740793784564520854062204...
%F A308962 a(n) ~ c * d^n * n^(4*n), where
%F A308962 d = 2.31219720619339615667811172118287009649702081583503593066663730992576726...
%F A308962 c = 17.2806567085831933774093124549232969200598807738253988225436890867215712...
%F A308962 (End)
%e A308962 a(1) = 17: (124)(3), (134)(2), 14(3)(2), (2)(134), (2)14(3), (234)(1), 24(3)(1), (3)(124), (3)14(2), (3)(2)14, (3)24(1), 34(2)(1), (4)(123), (4)13(2), (4)(2)13, (4)23(1), (4)(3)12; (odd length runs are shown between parentheses).
%t A308962 Flatten[{1, Table[(4 n)! * Coefficient[Expand[Normal[Series[t^2/(1 - t*x - (1 - t^2)*E^(-t*x)), {x, 0, 4*n}, {t, 0, 2*n}]]], x^(4*n)*t^(2*n)], {n, 1, 10}]}] (* _Vaclav Kotesovec_, Jul 09 2019 *)
%Y A308962 Cf. A097591.
%K A308962 nonn
%O A308962 0,2
%A A308962 _Alois P. Heinz_, Jul 03 2019