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A302901 Number of permutations of [n] having exactly eight alternating descents. 2

%I

%S 7936,151853,2866632,46901985,758805360,12127342203,195951082944,

%T 3216832016019,53984412657360,928559550102410,16402837435610856,

%U 297876978668583126,5564330063809902240,106938416843133746250,2114402162990417017920,43002161983507383542010

%N Number of permutations of [n] having exactly eight alternating descents.

%C Index i is an alternating descent of permutation p if either i is odd and p(i) > p(i+1), or i is even and p(i) < p(i+1).

%H Alois P. Heinz, <a href="/A302901/b302901.txt">Table of n, a(n) for n = 9..400</a>

%H D. Chebikin, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r132">Variations on descents and inversions in permutations</a>, The Electronic J. of Combinatorics, 15 (2008), #R132.

%F a(n) ~ (4 - Pi)^8 * 2^(n + 5/2) * n^(n + 17/2) / (8! * Pi^(n + 17/2) * exp(n)). - _Vaclav Kotesovec_, Apr 29 2018

%F E.g.f.: ((x^8 - 24*x^7 + 280*x^6 - 2352*x^5 + 15120*x^4 - 73920*x^3 + 262080*x^2 - 604800*x + 80640*sin(x) + 322560)*cos(x)^4 - x*((x^7 - 8*x^6 + 56*x^5 - 336*x^4 + 1680*x^3 - 6720*x^2 + 20160*x - 40320)*sin(x) + 123*x^7 - 1488*x^6 + 8568*x^5 - 34944*x^4 + 105840*x^3 - 228480*x^2 + 302400*x - 120960)*cos(x)^3 + ((- 124*x^8 + 2456*x^7 - 22064*x^6 + 123984*x^5 - 483840*x^4 + 1310400*x^3 - 2257920*x^2 + 1854720*x - 645120)*sin(x) - 1136*x^8 + 20088*x^7 - 157472*x^6 + 722736*x^5 - 2116800*x^4 + 3971520*x^3 - 4515840*x^2 + 3144960*x - 967680)*cos(x)^2 + ((1012*x^8 - 13808*x^7 + 81872*x^6 - 282240*x^5 + 624960*x^4 - 846720*x^3 + 564480*x^2 - 161280*x)*sin(x) + 1508*x^8 - 21472*x^7 + 129808*x^6 - 423360*x^5 + 786240*x^4 - 846720*x^3 + 564480*x^2 - 161280*x)*cos(x) + (2520*x^8 - 40592*x^7 + 286048*x^6 - 1149120*x^5 + 2862720*x^4 - 4515840*x^3 + 4515840*x^2 - 2580480*x + 645120)*sin(x) + 2520*x^8 - 40048*x^7 + 278432*x^6 - 1108800*x^5 + 2782080*x^4 - 4515840*x^3 + 4515840*x^2 - 2580480*x + 645120)/(40320*cos(x)^5 + (- 40320*sin(x) + 201600)*cos(x)^4 + (161280*sin(x) - 322560)*cos(x)^3 + (483840*sin(x) - 806400)*cos(x)^2 + (- 322560*sin(x) + 322560)*cos(x) - 645120*sin(x) + 645120). - _Vaclav Kotesovec_, Apr 30 2018

%p b:= proc(u, o) option remember; series(`if`(u+o=0, 1,

%p add(b(o+j-1, u-j)*x, j=1..u)+

%p add(b(o-j, u-1+j), j=1..o)), x, 10)

%p end:

%p a:= n-> coeff(b(n, 0), x, 9):

%p seq(a(n), n=9..30);

%t nmax = 30; Drop[CoefficientList[Series[((x^8 - 24*x^7 + 280*x^6 - 2352*x^5 + 15120*x^4 - 73920*x^3 + 262080*x^2 - 604800*x + 80640*Sin[x] + 322560)*Cos[x]^4 - x*((x^7 - 8*x^6 + 56*x^5 - 336*x^4 + 1680*x^3 - 6720*x^2 + 20160*x - 40320)*Sin[x] + 123*x^7 - 1488*x^6 + 8568*x^5 - 34944*x^4 + 105840*x^3 - 228480*x^2 + 302400*x - 120960)*Cos[x]^3 + ((- 124*x^8 + 2456*x^7 - 22064*x^6 + 123984*x^5 - 483840*x^4 + 1310400*x^3 - 2257920*x^2 + 1854720*x - 645120)*Sin[x] - 1136*x^8 + 20088*x^7 - 157472*x^6 + 722736*x^5 - 2116800*x^4 + 3971520*x^3 - 4515840*x^2 + 3144960*x - 967680)*Cos[x]^2 + ((1012*x^8 - 13808*x^7 + 81872*x^6 - 282240*x^5 + 624960*x^4 - 846720*x^3 + 564480*x^2 - 161280*x)*Sin[x] + 1508*x^8 - 21472*x^7 + 129808*x^6 - 423360*x^5 + 786240*x^4 - 846720*x^3 + 564480*x^2 - 161280*x)*Cos[x] + (2520*x^8 - 40592*x^7 + 286048*x^6 - 1149120*x^5 + 2862720*x^4 - 4515840*x^3 + 4515840*x^2 - 2580480*x + 645120)*Sin[x] + 2520*x^8 - 40048*x^7 + 278432*x^6 - 1108800*x^5 + 2782080*x^4 - 4515840*x^3 + 4515840*x^2 - 2580480*x + 645120)/(40320*Cos[x]^5 + (- 40320*Sin[x] + 201600)*Cos[x]^4 + (161280*Sin[x] - 322560)*Cos[x]^3 + (483840*Sin[x] - 806400)*Cos[x]^2 + (- 322560*Sin[x] + 322560)*Cos[x] - 645120*Sin[x] + 645120), {x, 0, nmax}], x] * Range[0, nmax]!, 9] (* _Vaclav Kotesovec_, Apr 30 2018 *)

%Y Column k=9 of A145876.

%K nonn

%O 9,1

%A _Alois P. Heinz_, Apr 15 2018

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Last modified May 27 11:47 EDT 2022. Contains 354097 sequences. (Running on oeis4.)