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A000274 Number of permutations of length n with 2 consecutive ascending pairs.
(Formerly M3048 N1236)
9

%I M3048 N1236

%S 0,0,1,3,18,110,795,6489,59332,600732,6674805,80765135,1057289046,

%T 14890154058,224497707343,3607998868005,61576514013960,

%U 1112225784377144,21197714949305577,425131949816628507,8950146311929021210

%N Number of permutations of length n with 2 consecutive ascending pairs.

%C From _Emeric Deutsch_, May 25 2009: (Start)

%C a(n) = number of excedances in all derangements of [n-1]. Example: a(5)=18 because the derangements of {1,2,3,4} are 4*123, 3*14*2, 3*4*12, 4*3*12, 2*14*3, 2*4*13, 2*3*4*1, 3*4*21, 4*3*21 with the 18 excedances marked. An excedance of a permutation p is a position i such that p(i)>i.

%C a(n) = Sum(k*A046739(n,k), k>=1).

%C (End)

%C Appears to be the inverse binomial transform of A001286 (filling the two leading zeros in there), then shifting one place to the right. _R. J. Mathar_, Apr 04 2012

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

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210 (divided by 2).

%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 Alois P. Heinz, <a href="/A000274/b000274.txt">Table of n, a(n) for n = 1..150</a>

%H R. Mantaci and F. Rakotondrajao, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00531-6">Exceedingly deranging!</a>, Advances in Appl. Math., 30 (2003), 177-188. [_Emeric Deutsch_, May 25 2009]

%F a(n) = (1 + n) a(n - 1) + (3 + n) a(n - 2) + (3 - n) a(n - 3) + (2 - n) a(n - 4).

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

%F a(n) = (n-1)^2/(n-2)*a(n-1)-(-1)^n*(n-1)/2, n>2, a(2)=0. - _Vladeta Jovovic_, Aug 31 2003

%F a(n) = (1/2){[n!/e] - [(n-1)!/e]} (conjectured).

%F a(n) = (n-1)*GAMMA(n,-1)*exp(-1)/2 where GAMMA = incomplete Gamma function. [_Mark van Hoeij_, Nov 11 2009]

%F a(n) = A145887(n-1) + A145886(n-1). - _Anton Zakharov_, Aug 28 2016

%p a:= n->sum(n!*sum((-1)^k/k!/2, j=1..n), k=0..n): seq(a(n), n=2..20); # _Zerinvary Lajos_, May 17 2007

%t Table[Subfactorial[n]*n/2, {n, 2, 20}] (* _Zerinvary Lajos_, Jul 09 2009 *)

%Y Cf. A010027, A000255, A000166, A000313, A001260, A001261.

%Y A diagonal in triangle A010027.

%Y Cf. A046739. [_Emeric Deutsch_, May 25 2009]

%Y Cf. A145887, A145886.

%K easy,nonn

%O 1,4

%A _N. J. A. Sloane_, _Simon Plouffe_

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

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Last modified February 21 01:09 EST 2019. Contains 320364 sequences. (Running on oeis4.)