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 A049376 #55 Jan 18 2025 09:05:27
%S A049376 1,1,4,22,154,1306,12976,147484,1883932,26680924,414468496,7001104936,
%T A049376 127677078904,2498712779512,52209534323584,1159559538626896,
%U A049376 27269218041047056,676732851527182864,17669429275516846912,484087943980439097184,13882791112964223876256
%N A049376 Row sums of triangle A046089.
%C A049376 a(n) is the number of n-permutations where each cycle has two (not necessarily distinct) roots. Here a root means a designated element in a cycle. Cf. A000262 which gives the number of n-permutations with a single root in each cycle. Note that the order of designating the elements is not important. Cf. (A bijection from endofunctions to "doubly" rooted trees where the order of designating the roots is important) Miklos Bona, A Walk Through Combinatorics, World Scientific Publishing, 2006, page 216. - _Geoffrey Critzer_, May 17 2012.
%H A049376 Alois P. Heinz, <a href="/A049376/b049376.txt">Table of n, a(n) for n = 0..436</a>
%H A049376 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F A049376 E.g.f.: exp(p(x)) with p(x) := x*(2-x)/(2*(1-x)^2) (E.g.f. first column of A046089).
%F A049376 Lah transform of A000085: a(n) = Sum_{k=0..n} n!/k!*binomial(n-1,k-1) * A000085(k). - _Vladeta Jovovic_, Oct 02 2003
%F A049376 a(n+3) - (3*n+7)*a(n+2) + 3*(n+1)*(n+2)*a(n+1) - n*(n+1)*(n+2)* a(n) = 0. - _Emanuele Munarini_, Sep 08 2017
%F A049376 a(n) ~ n^(n-1/6) / sqrt(3) * exp(-1/3 + n^(1/3)/2 + 3*n^(2/3)/2 - n). - _Vaclav Kotesovec_, Oct 23 2017
%F A049376 E.g.f.: Sum_{n>=0} ( Integral 1/(1-x)^3 dx )^n / n!, where the constant of integration is taken to be zero. - _Paul D. Hanna_, Apr 27 2019
%F A049376 From _Seiichi Manyama_, Jan 18 2025: (Start)
%F A049376 a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A004211(k).
%F A049376 a(n) = (1/exp(1/2)) * (-1)^n * n! * Sum_{k>=0} binomial(-2*k,n)/(2^k * k!). (End)
%e A049376 a(2) = 4 because we have: (1'')(2'');(1''2);(12'');(1'2') where the permutations are given in cycle notation and the two roots in each cycle are designated by a '.
%p A049376 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A049376 binomial(n-1, j-1)*(j+1)!/2*a(n-j), j=1..n))
%p A049376 end:
%p A049376 seq(a(n), n=0..25); # _Alois P. Heinz_, Aug 01 2017
%p A049376 a := proc(n) option remember; `if`(n < 3, [1, 1, 4][n + 1],
%p A049376 a(n-1)*(3*n-2) - a(n-2)*3*(n-1)*(n-2) + a(n-3)*(n-1)*(n-2)*(n-3)) end:
%p A049376 seq(a(n), n=0..20); # after _Emanuele Munarini_, _Peter Luschny_, Sep 09 2017
%t A049376 nn = 15;Drop[Range[0, nn]! CoefficientList[Series[Exp[x/(1 - x) + x^2/2/(1 - x)^2], {x, 0, nn}], x], 1] (* _Geoffrey Critzer_, May 17 2012 *)
%Y A049376 Column k=3 of A291709.
%Y A049376 Cf. A000085, A000262, A046089, A136658.
%Y A049376 Cf. A004211, A375173.
%K A049376 nonn
%O A049376 0,3
%A A049376 _Wolfdieter Lang_
%E A049376 a(0)=1 prepended by _Alois P. Heinz_, Aug 01 2017