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A128103 Number of permutations of [n] with an even number of rises.

%I A128103 #25 Jan 24 2020 15:45:12
%S A128103 1,1,1,2,12,68,360,2384,20160,185408,1814400,19781504,239500800,
%T A128103 3124694528,43589145600,652885305344,10461394944000,177948646719488,
%U A128103 3201186852864000,60808005761859584,1216451004088320000,25547946834881282048,562000363888803840000
%N A128103 Number of permutations of [n] with an even number of rises.
%H A128103 Alois P. Heinz, <a href="/A128103/b128103.txt">Table of n, a(n) for n = 0..400</a>
%H A128103 F. C. S. Brown, T. M. A. Fink and K. Willbrand, <a href="https://arxiv.org/abs/math/0607763">On arithmetic and asymptotic properties of up-down numbers</a>, arXiv:math/0607763 [math.CO], 2006.
%F A128103 E.g.f.: 1 + 1/2 [z/(1-z) + tanh(z) ].
%F A128103 a(n) = A000142(n) - A262745(n).
%F A128103 If n is even, a(n) = (n)!/2 (A002674), if n is odd, a(n) = (n)! * (1 + (-1)^((n-1)/2) * A002430((n+1)/2) / A036279((n+1)/2)) / 2. - _Michel Marcus_, Dec 09 2012
%F A128103 Conjecture: a(n) = Sum_{k = 0..n} Sum_{j = 0..k} (-1)^(n+j)*binomial(n,k-j)*j^n. - _Peter Bala_, Jan 22 2020
%p A128103 b:= proc(u, o, t) option remember; `if`(u+o=0, t,
%p A128103       add(b(u-j, o+j-1, t), j=1..u)+
%p A128103       add(b(u+j-1, o-j, 1-t), j=1..o))
%p A128103     end:
%p A128103 a:= n-> b(n, 0, 1):
%p A128103 seq(a(n), n=0..25);  # _Alois P. Heinz_, Sep 29 2015
%t A128103 b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, t, Sum[b[u - j, o + j - 1, t], {j, 1, u}] + Sum[b[u + j - 1, o - j, 1 - t], {j, 1, o}]];
%t A128103 a[n_] := b[n, 0, 1];
%t A128103 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jul 25 2017, after _Alois P. Heinz_ *)
%o A128103 (PARI) x='x+O('x^99); Vec(serlaplace((x/(1-x)+tanh(x))/2+1)) \\ _Altug Alkan_, Jul 25 2017
%Y A128103 Cf. A000142, A002674, A002430, A036279, A262745.
%K A128103 nonn
%O A128103 0,4
%A A128103 _Ralf Stephan_, May 09 2007
%E A128103 More terms from _Alois P. Heinz_, Sep 29 2015