%I M3999 N1657 #43 May 09 2020 15:11:53

%S 0,0,0,0,1,5,45,385,3710,38934,444990,5506710,73422855,1049946755,

%T 16035550531,260577696015,4489954146860,81781307674780,

%U 1570201107355980,31698434854748604,671260973394676605,14879618243581997745

%N Number of permutations of length n with 4 consecutive ascending pairs.

%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 Vincenzo Librandi, <a href="/A001260/b001260.txt">Table of n, a(n) for n = 1..200</a>

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

%F E.g.f.: (for offset 4): (x^4/4!)*exp(-x)/(1-x)^2. - _Vladeta Jovovic_, Jan 03 2003

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

%F 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

%F a(n) ~ n! * exp(-1)/24. - _Vaclav Kotesovec_, Mar 26 2014

%p 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

%p series(hypergeom([2, 5],[],x/(x+1))/(x+1)^5,x=0,30); # _Mark van Hoeij_, Nov 07 2011

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

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

%Y A diagonal in triangle A010027.

%K nonn

%O 1,6

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, Jan 03 2003

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