A302900 Number of permutations of [n] having exactly seven alternating descents.

1385, 21682, 350240, 4945368, 70145634, 992272308, 14322115212, 211595659320, 3216832016019, 50412205403030, 815486339550108, 13622914005990480, 235041722344009380, 4187522527966916520, 77010173788311008040, 1461190162226869057872, 28588437379997078589117

Offset

8,1

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 = 8..400

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

Formula

a(n) ~ (4 - Pi)^7 * 2^(n + 5/2) * n^(n + 15/2) / (7! * Pi^(n + 15/2) * exp(n)). - Vaclav Kotesovec, Apr 29 2018

E.g.f.: (5040*cos(x)^4 + (7*x^6 - 84*x^5 + 630*x^4 - 3360*x^3 + 12600*x^2 - 30240*x + 5040*sin(x) + 15120)*cos(x)^3 + ((x^7 - 14*x^6 + 126*x^5 - 840*x^4 + 4200*x^3 - 15120*x^2 + 35280*x - 20160)*sin(x) + 60*x^7 - 840*x^6 + 5544*x^5 - 23520*x^4 + 67200*x^3 - 120960*x^2 + 105840*x - 40320)*cos(x)^2 + ((- 196*x^6 + 2352*x^5 - 12600*x^4 + 40320*x^3 - 75600*x^2 + 60480*x - 20160)*sin(x) - 434*x^6 + 5208*x^5 - 25200*x^4 + 60480*x^3 - 75600*x^2 + 60480*x - 20160)*cos(x) + (- 298*x^7 + 4172*x^6 - 25200*x^5 + 85680*x^4 - 176400*x^3 + 211680*x^2 - 141120*x + 40320)*sin(x) - 332*x^7 + 4648*x^6 - 27720*x^5 + 90720*x^4 - 176400*x^3 + 211680*x^2 - 141120*x + 40320)/(5040*cos(x)^4 + (20160*sin(x) - 40320)*cos(x)^2 - 40320*sin(x) + 40320). - 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, 9)
    end:
a:= n-> coeff(b(n, 0), x, 8):
seq(a(n), n=8..30);

Mathematica

nmax = 30; Drop[CoefficientList[Series[(5040*Cos[x]^4 + (7*x^6 - 84*x^5 + 630*x^4 - 3360*x^3 + 12600*x^2 - 30240*x + 5040*Sin[x] + 15120)*Cos[x]^3 + ((x^7 - 14*x^6 + 126*x^5 - 840*x^4 + 4200*x^3 - 15120*x^2 + 35280*x - 20160)*Sin[x] + 60*x^7 - 840*x^6 + 5544*x^5 - 23520*x^4 + 67200*x^3 - 120960*x^2 + 105840*x - 40320)*Cos[x]^2 + ((- 196*x^6 + 2352*x^5 - 12600*x^4 + 40320*x^3 - 75600*x^2 + 60480*x - 20160)*Sin[x] - 434*x^6 + 5208*x^5 - 25200*x^4 + 60480*x^3 - 75600*x^2 + 60480*x - 20160)*Cos[x] + (- 298*x^7 + 4172*x^6 - 25200*x^5 + 85680*x^4 - 176400*x^3 + 211680*x^2 - 141120*x + 40320)*Sin[x] - 332*x^7 + 4648*x^6 - 27720*x^5 + 90720*x^4 - 176400*x^3 + 211680*x^2 - 141120*x + 40320)/(5040*Cos[x]^4 + (20160*Sin[x] - 40320)*Cos[x]^2 - 40320*Sin[x] + 40320), {x, 0, nmax}], x] * Range[0, nmax]!, 8] (* Vaclav Kotesovec, Apr 30 2018 *)

Crossrefs

Column k=8 of A145876.

Sequence in context: A336192 A345526 A345780 * A317284 A060062 A181991

Adjacent sequences: A302897 A302898 A302899 * A302901 A302902 A302903

Keyword

nonn

Author

Alois P. Heinz, Apr 15 2018

Status

approved