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

7936, 151853, 2866632, 46901985, 758805360, 12127342203, 195951082944, 3216832016019, 53984412657360, 928559550102410, 16402837435610856, 297876978668583126, 5564330063809902240, 106938416843133746250, 2114402162990417017920, 43002161983507383542010

Offset

9,1

Comments

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).

Links

Alois P. Heinz, Table of n, a(n) for n = 9..400

D. Chebikin, Variations on descents and inversions in permutations, The Electronic J. of Combinatorics, 15 (2008), #R132.

Formula

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

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

Maple

b:= proc(u, o) option remember; series(`if`(u+o=0, 1,
       add(b(o+j-1, u-j)*x, j=1..u)+
       add(b(o-j, u-1+j),   j=1..o)), x, 10)
    end:
a:= n-> coeff(b(n, 0), x, 9):
seq(a(n), n=9..30);

Mathematica

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 *)

Crossrefs

Column k=9 of A145876.

Sequence in context: A250629 A185466 A028537 * A179708 A162010 A023322

Adjacent sequences: A302898 A302899 A302900 * A302902 A302903 A302904

Keyword

nonn

Author

Alois P. Heinz, Apr 15 2018

Status

approved