This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A000274 M3048 N1236 #43 Nov 26 2024 07:01:30
%S A000274 0,0,1,3,18,110,795,6489,59332,600732,6674805,80765135,1057289046,
%T A000274 14890154058,224497707343,3607998868005,61576514013960,
%U A000274 1112225784377144,21197714949305577,425131949816628507,8950146311929021210,197350726178034917670,4548464355722328578691
%N A000274 Number of permutations of length n with 2 consecutive ascending pairs.
%C A000274 From _Emeric Deutsch_, May 25 2009: (Start)
%C A000274 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 A000274 a(n) = Sum(k*A046739(n,k), k>=1).
%C A000274 (End)
%C A000274 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 A000274 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
%D A000274 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210 (divided by 2).
%D A000274 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000274 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000274 Alois P. Heinz, <a href="/A000274/b000274.txt">Table of n, a(n) for n = 1..150</a>
%H A000274 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 A000274 a(n) = (1 + n) a(n - 1) + (3 + n) a(n - 2) + (3 - n) a(n - 3) + (2 - n) a(n - 4).
%F A000274 E.g.f.: x^2/2*exp(-x)/(1-x)^2. - _Vladeta Jovovic_, Jan 03 2003
%F A000274 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 A000274 a(n) = (1/2){[n!/e] - [(n-1)!/e]} (conjectured).
%F A000274 a(n) = (n-1)*GAMMA(n,-1)*exp(-1)/2 where GAMMA = incomplete Gamma function. [_Mark van Hoeij_, Nov 11 2009]
%F A000274 a(n) = A145887(n-1) + A145886(n-1). - _Anton Zakharov_, Aug 28 2016
%p A000274 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
%t A000274 Table[Subfactorial[n]*n/2, {n, 2, 20}] (* _Zerinvary Lajos_, Jul 09 2009 *)
%Y A000274 Cf. A010027, A000255, A000166, A000313, A001260, A001261.
%Y A000274 A diagonal in triangle A010027.
%Y A000274 Cf. A046739. [_Emeric Deutsch_, May 25 2009]
%Y A000274 Cf. A145887, A145886.
%K A000274 easy,nonn
%O A000274 1,4
%A A000274 _N. J. A. Sloane_, _Simon Plouffe_
%E A000274 Name clarified and offset changed by _N. J. A. Sloane_, Apr 12 2014