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

0, 0, 1, 3, 18, 110, 795, 6489, 59332, 600732, 6674805, 80765135, 1057289046, 14890154058, 224497707343, 3607998868005, 61576514013960, 1112225784377144, 21197714949305577, 425131949816628507, 8950146311929021210, 197350726178034917670, 4548464355722328578691

Offset

1,4

Comments

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. See Mantaci and Rakotondrajao article. - Emeric Deutsch, May 25 2009

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

Mathar's conjecture is true and the conjectured formula a(n) = (1/2)*([n!/e] - [(n-1)!/e]) is true (see Fried link). - Sela Fried, Sep 27 2025

References

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

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

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

Alois P. Heinz, Table of n, a(n) for n = 1..150

Sela Fried, On sequence A000274 in the OEIS, 2025.

Roberto Mantaci and Fanja Rakotondrajao, Exceedingly deranging!, Advances in Appl. Math., 30 (2003), 177-188.

Formula

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

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

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

a(n) = (1/2)*([n!/e] - [(n-1)!/e]) (conjectured).

a(n) = Sum_{k>=1} (k*A046739(n,k)). - Emeric Deutsch, May 25 2009

a(n) = (n-1)*Gamma(n,-1)*exp(-1)/2 where Gamma = incomplete Gamma function. - Mark van Hoeij, Nov 11 2009

a(n) = A145887(n-1) + A145886(n-1). - Anton Zakharov, Aug 28 2016

a(n) ~ sqrt(Pi/2) * n^(n+1/2) / exp(n+1). - Amiram Eldar, Sep 20 2025 [corrected by Sela Fried, Sep 27 2025]

Maple

a:= n->sum((n-1)!*sum((-1)^k/k!/2, j=1..n-1), k=0..n-1): seq(a(n), n=1..23); # Zerinvary Lajos, May 17 2007

Mathematica

Table[Subfactorial[n]*n/2, {n, 2, 20}] (* Zerinvary Lajos, Jul 09 2009 *)

Crossrefs

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

A diagonal in triangle A010027.

Cf. A046739. - Emeric Deutsch, May 25 2009

Cf. A145887, A145886.

Sequence in context: A074571 A114311 A134092 * A207321 A193236 A357203

Adjacent sequences: A000271 A000272 A000273 * A000275 A000276 A000277

Keyword

easy,nonn

Author

N. J. A. Sloane and Simon Plouffe

Extensions

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

Status

approved