A395484 Number of derangements of [n] with 5 descents.

0, 0, 0, 0, 0, 0, 1, 54, 1764, 34520, 495320, 5794944, 58753904, 536362784, 4522914840, 35866688640, 270967688256, 1969261200000, 13869344791528, 95206279619680, 639874975302480, 4225823731044064, 27502902905426872, 176817788582956224, 1125118788057967200

Offset

0,8

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (50, -1155, 16352, -158683, 1118106, -5909837, 23861764, -74208228, 177768976, -325379888, 446917824, -445952448, 305227008, -128148480, 24883200).

Formula

E.g.f.: 15625/46656 * exp(6*x) - 1024*(1+5*x)/3125 * exp(5*x) + 81*x*(1+2*x)/64 * exp(4*x) - 4*x^2*(1+x)/3 * exp(3*x) + x^3*(2+x)/6 * exp(2*x) - 1052381/145800000 + 17371*x/9720000 - 47*x^2/81000 + 11*x^3/32400 - x^4/864 - x^5/720.

G.f.: x^6 * (1+4*x+219*x^2-7662*x^3+82415*x^4-494608*x^5+2079189*x^6-6977126*x^7+19080352*x^8-40144336*x^9+61283792*x^10-65556960*x^11+47260800*x^12-20666880*x^13+4147200*x^14)/((1-2*x)^5 * (1-3*x)^4 * (1-4*x)^3 * (1-5*x)^2 * (1-6*x)).

a(n) = 50*a(n-1) - 1155*a(n-2) + 16352*a(n-3) - 158683*a(n-4) + 1118106*a(n-5) - 5909837*a(n-6) + 23861764*a(n-7) - 74208228*a(n-8) + 177768976*a(n-9) - 325379888*a(n-10) + 446917824*a(n-11) - 445952448*a(n-12) + 305227008*a(n-13) - 128148480*a(n-14) + 24883200*a(n-15) for n > 19.

Prog

(PARI) my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0], Vec(x^6*(1+4*x+219*x^2-7662*x^3+82415*x^4-494608*x^5+2079189*x^6-6977126*x^7+19080352*x^8-40144336*x^9+61283792*x^10-65556960*x^11+47260800*x^12-20666880*x^13+4147200*x^14)/((1-2*x)^5*(1-3*x)^4*(1-4*x)^3*(1-5*x)^2*(1-6*x))))

Crossrefs

Column k=5 of A219836.

Sequence in context: A017717 A172501 A173188 * A004363 A062144 A262112

Adjacent sequences: A395481 A395482 A395483 * A395485 A395486 A395487

Keyword

nonn

Author

Seiichi Manyama, Apr 25 2026

Status

approved