A000313 - OEIS

%I M3633 N1477 #78 Jul 28 2024 16:39:39

%S 0,0,0,1,4,30,220,1855,17304,177996,2002440,24474285,323060540,

%T 4581585866,69487385604,1122488536715,19242660629360,348933579412440,

%U 6673354706262864,134252194678935321,2834212998777523380,62651024183503148470,1447238658638922729580

%N Number of permutations of length n with 3 consecutive ascending pairs.

%C 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

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

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

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

%H Todd Silvestri, <a href="/A000313/b000313.txt">Table of n, a(n) for n = 1..450</a> (first 100 terms from T. D. Noe)

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

%F a(n) = A065087(n+2)/3. - _Zerinvary Lajos_, May 25 2007

%F E.g.f.: x^3/3!*exp(-x)/(1-x)^2. - _Vladeta Jovovic_, Jan 03 2003

%F a(n) = round( (exp(-1)*(n+1)!+(-1)^n)*n/6 ). - _Mark van Hoeij_, Oct 25 2011

%F G.f.: hypergeom([2, 4],[],x/(x+1))/(x+1)^4. - _Mark van Hoeij_, Nov 07 2011

%F 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

%F a(n) = hypergeom([4-n,2],[],1)*(-1)^n*A000292(n-3). - _Peter Luschny_, Nov 19 2014

%F 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

%p series(hypergeom([2,4],[],x/(x+1))/(x+1)^4, x=0, 30); # _Mark van Hoeij_, Nov 07 2011

%p 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

%t Table[(n*(n + 1)!/6)*Sum[(-1)^k/k!, {k, 0, n}], {n, -1, 25}] (* _T. D. Noe_, Jun 19 2012 *)

%t a[1]:=0; a[n_Integer/;n>=2]:=(n-2) (n-1) Subfactorial[n-2]/6 (* _Todd Silvestri_, Nov 15 2014 *)

%o (Sage)

%o a = lambda n: (n-2)*(n-1)*sloane.A000166(n-2)/6 if n>2 else 0

%o [a(n) for n in range(1,24)] # _Peter Luschny_, Nov 19 2014

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

%Y A diagonal in triangle A010027.

%K nonn,easy

%O 1,5

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, Jan 03 2003

%E Formula added by _Sean A. Irvine_, Nov 11 2010

%E Name clarified and offset changed by _N. J. A. Sloane_, Apr 12 2014