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 A088436 #46 Sep 08 2022 08:45:11
%S A088436 0,1,3,6,30,225,1575,12180,109620,1100925,12110175,145259730,
%T A088436 1888376490,26438216805,396573252075,6345155817000,107867648889000,
%U A088436 1941617990136825,36890741812599675,737814829704702750
%N A088436 Number of permutations in the symmetric group S_n that have exactly one transposition in their cycle decomposition.
%D A088436 Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 189, Exercise 19 for k=1. With (-1)^k omitted.
%H A088436 Muniru A Asiru, <a href="/A088436/b088436.txt">Table of n, a(n) for n = 1..440</a>
%F A088436 a(n) = (n!/2)*Sum_{j=0..floor(n/2)-1} (-1)^j/(j!*2^j), n >= 1.
%F A088436 E.g.f.: x^2/(1-x)/2*exp(-x^2/2). - _Vladeta Jovovic_, Nov 09 2003
%F A088436 From _Paul Weisenhorn_, Jun 02 2010: (Start)
%F A088436 In general, for k cycles of length 2,
%F A088436 a(n) = n!*Sum_{j=k..floor(n/2)} (-1)^j/((j-k)!*2^j*k!).
%F A088436 G.f.: (exp(-z^2/2)*z^2*k)/((1-z)*2^k*k!). (End)
%F A088436 a(n) ~ exp(-1/2)/2 * n!. - _Vaclav Kotesovec_, Mar 18 2014
%e A088436 From _Bernard Schott_, Feb 19 2019: (Start)
%e A088436 For S_4, the six permutations that have exactly one transposition in their cycle decomposition are (12)(3)(4), (13)(2)(4), (14)(2)(3), (23)(1)(4), (24)(1)(3), (34)(1)(2).
%e A088436 For S_5, there are exactly 10 transpositions: (12), (13), (14), (15), (23), (24), (25), (34), (35), (45), and for each transposition, there are 3 permutations that have exactly this transposition and no other transposition in their cycle decomposition; for example, for transposition (12), these three permutations: (12)(3)(4)(5), (12)(345), (12)(354), so a(5) = 10 * 3 = 30. (End)
%p A088436 G=(exp(-z^2/2)*z^2*k)/((1-z)*2^k*k!): Gser=series(G,z=0,21):
%p A088436 for n from 2*k to 20 do a(n)=n!*coeff(Gser,z,n): end do: # _Paul Weisenhorn_, Jun 02 2010
%t A088436 d=Exp[-x^2/2]/(1-x); Range[0,20]! CoefficientList[Series[(x^2/2! )d, {x,0,20}], x] (* _Geoffrey Critzer_, Nov 29 2011 *)
%o A088436 (PARI) my(x='x+O('x^30)); concat([0], Vec(serlaplace( x^2*exp(-x^2/2)/(2*(1-x)) ))) \\ _G. C. Greubel_, Feb 19 2019
%o A088436 (Magma) m:=32; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( x^2*Exp(-x^2/2)/(2*(1-x)) )); [0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // _G. C. Greubel_, Feb 19 2019
%o A088436 (Sage) m = 30; T = taylor(x^2*exp(-x^2/2)/(2*(1-x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (1..m)] # _G. C. Greubel_, Feb 19 2019
%Y A088436 Cf. A000266, A027616, A000240.
%K A088436 nonn
%O A088436 1,3
%A A088436 Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 09 2003
%E A088436 More terms from _Wolfdieter Lang_, Feb 22 2008