A049376 Row sums of triangle A046089.

1, 1, 4, 22, 154, 1306, 12976, 147484, 1883932, 26680924, 414468496, 7001104936, 127677078904, 2498712779512, 52209534323584, 1159559538626896, 27269218041047056, 676732851527182864, 17669429275516846912, 484087943980439097184, 13882791112964223876256

Offset

0,3

Comments

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.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..436

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Formula

E.g.f.: exp(p(x)) with p(x) := x*(2-x)/(2*(1-x)^2) (E.g.f. first column of A046089).

Lah transform of A000085: a(n) = Sum_{k=0..n} n!/k!*binomial(n-1,k-1) * A000085(k). - Vladeta Jovovic, Oct 02 2003

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

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

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

Example

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 '.

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+1)!/2*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017
a := proc(n) option remember; `if`(n < 3, [1, 1, 4][n + 1],
a(n-1)*(3*n-2) - a(n-2)*3*(n-1)*(n-2) + a(n-3)*(n-1)*(n-2)*(n-3)) end:
seq(a(n), n=0..20); # after Emanuele Munarini, Peter Luschny, Sep 09 2017

Mathematica

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 *)

Crossrefs

Cf. A000085, A000262, A046089, A136658.

Sequence in context: A196275 A000307 A294346 * A083410 A295553 A302548

Adjacent sequences: A049373 A049374 A049375 * A049377 A049378 A049379

Keyword

nonn

Author

Wolfdieter Lang

Extensions

a(0)=1 prepended by Alois P. Heinz, Aug 01 2017

Status

approved