A001260 Number of permutations of length n with 4 consecutive ascending pairs. (Formerly M3999 N1657)

0, 0, 0, 0, 1, 5, 45, 385, 3710, 38934, 444990, 5506710, 73422855, 1049946755, 16035550531, 260577696015, 4489954146860, 81781307674780, 1570201107355980, 31698434854748604, 671260973394676605, 14879618243581997745

Offset

1,6

References

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Formula

(n-1)*a(n) = (n+3)*(a(n-1)*n + a(n-2)*n - a(n-1) + 2*a(n-2)).

E.g.f.: (for offset 4): (x^4/4!)*exp(-x)/(1-x)^2. - Vladeta Jovovic, Jan 03 2003

G.f.: (for offset 0): hypergeom([2, 5],[],x/(x+1))/(x+1)^5. - Mark van Hoeij, Nov 07 2011

Recurrence (for offset 5): (n-5)*a(n) = (n-5)*(n-1)*a(n-1) + (n-2)*(n-1)*a(n-2). - Vaclav Kotesovec, Mar 26 2014

a(n) ~ n! * exp(-1)/24. - Vaclav Kotesovec, Mar 26 2014

Maple

a:=n->sum((n+2)!*sum((-1)^k/k!/4!, j=1..n), k=0..n): seq(a(n), n=2..19); # Zerinvary Lajos, May 25 2007
series(hypergeom([2, 5], [], x/(x+1))/(x+1)^5, x=0, 30); # Mark van Hoeij, Nov 07 2011

Mathematica

Drop[CoefficientList[Series[x^4/4! Exp[-x]/(1 - x)^2, {x, 0, 20}], x] Range[0, 20]!, 4] (* Vaclav Kotesovec, Mar 26 2014 *)

Crossrefs

Cf. A010027, A000255, A000166, A000274, A000313, A001261.

A diagonal in triangle A010027.

Sequence in context: A272494 A185009 A125836 * A088505 A067403 A173292

Adjacent sequences: A001257 A001258 A001259 * A001261 A001262 A001263

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Jan 03 2003

Name clarified and offset changed by N. J. A. Sloane, Apr 12 2014

Status

approved