A000313 Number of permutations of length n with 3 consecutive ascending pairs.

0, 0, 0, 1, 4, 30, 220, 1855, 17304, 177996, 2002440, 24474285, 323060540, 4581585866, 69487385604, 1122488536715, 19242660629360, 348933579412440, 6673354706262864, 134252194678935321, 2834212998777523380, 62651024183503148470, 1447238658638922729580

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Temporary remark: there may be some issues with respect to the offset of this sequence in the formula and program sections. - Joerg Arndt, Nov 16 2014

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

• Todd Silvestri, Table of n, a(n) for n = 1..450 (first 100 terms from T. D. Noe)

Crossrefs

Cf. A010027, A000255, A000166, A000274, A001260, A001261.

A diagonal in triangle A010027.

Programs

Maple

series(hypergeom([2,4],[],x/(x+1))/(x+1)^4, x=0, 30); # Mark van Hoeij, Nov 07 2011
a := n -> simplify(hypergeom([4-n,2],[],1))*(-1)^n*(n-1)*(n-2)*(n-3)/6: seq(a(n), n=1..23); # Peter Luschny, Nov 19 2014

Mathematica

Table[(n*(n + 1)!/6)*Sum[(-1)^k/k!, {k, 0, n}], {n, -1, 25}] (* T. D. Noe, Jun 19 2012 *)
a[1]:=0; a[n_Integer/;n>=2]:=(n-2) (n-1) Subfactorial[n-2]/6 (* Todd Silvestri, Nov 15 2014 *)

Sage

a = lambda n: (n-2)*(n-1)*sloane.A000166(n-2)/6 if n>2 else 0
[a(n) for n in range(1,24)] # Peter Luschny, Nov 19 2014

Formula

a(n) = (n*(n+1)!/6)*sum((-1)^k/k!, k=0..n).

a(n) = A065087(n+2)/3. - Zerinvary Lajos, May 25 2007

E.g.f.: x^3/3!*exp(-x)/(1-x)^2. - Vladeta Jovovic, Jan 03 2003

a(n) = round( (exp(-1)*(n+1)!+(-1)^n)*n/6 ). - Mark van Hoeij, Oct 25 2011

G.f.: hypergeom([2, 4],[],x/(x+1))/(x+1)^4. - Mark van Hoeij, Nov 07 2011

a(1) = 0, a(n) = (n-2)*(n-1)*(!(n-2))/6 = (n-2)*(n-1)*A000166(n-2)/6, for n >= 2. - Todd Silvestri, Nov 15 2014

a(n) = hypergeom([4-n,2],[],1)*(-1)^n*A000292(n-3). - Peter Luschny, Nov 19 2014

D-finite with recurrence (-n+4)*a(n) +(n-1)*(n-4)*a(n-1) +(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Aug 01 2022

Extensions

More terms from Vladeta Jovovic, Jan 03 2003

Formula added by Sean A. Irvine, Nov 11 2010

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