login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A106239 Triangle read by rows: T(n,m) = number of graphs on n labeled nodes, with m components of size = order. Also number of graphs on n labeled nodes with m unicyclic components. 4
0, 0, 0, 1, 0, 0, 15, 0, 0, 0, 222, 0, 0, 0, 0, 3660, 10, 0, 0, 0, 0, 68295, 525, 0, 0, 0, 0, 0, 1436568, 20307, 0, 0, 0, 0, 0, 0, 33779340, 727020, 280, 0, 0, 0, 0, 0, 0, 880107840, 25934184, 31500, 0, 0, 0, 0, 0, 0, 0, 25201854045, 950478210, 2325015, 0, 0, 0, 0, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,7

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

Washington Bomfim, Illustration of this sequence

FORMULA

E.g.f.: exp(-y/2*ln(1+LambertW(-x))+y/2*LambertW(-x)-y/4*LambertW(-x)^2). - Vladeta Jovovic, May 04 2005

T(n,m) = sum N/D over the partitions of n:1K1+2K2+ ... +nKn, = with exactly m parts greater than 2, where N = n!*product_{1=<i<=n}= A057500(i)^Ki and D = product_{1=<i<=n}(Ki!(i!)^Ki).

T(n,1) = A057500(n), T(n,m) = Sum_{j=2..n-1} C(n-1,j) * A057500(j+1) * T(n-1-j,m-1) if m>1. - Alois P. Heinz, Sep 15 2008

EXAMPLE

T(6,2) = 10 because there are 10 such graphs of order 6 with 2 components. The value of T(n,m) is zero if and only if m > floor(n/3).

Triangle T(n,m) begins:

0;

0, 0;

1, 0, 0;

15, 0, 0, 0;

222, 0, 0, 0, 0;

3660, 10, 0, 0, 0, 0;

68295, 525, 0, 0, 0, 0, 0;

1436568, 20307, 0, 0, 0, 0, 0, 0;

33779340, 727020, 280, 0, 0, 0, 0, 0, 0;

880107840, 25934184, 31500, 0, 0, 0, 0, 0, 0, 0;

MAPLE

cy:= proc(n) option remember; local t; binomial(n-1, 2) *add ((n-3)! /(n-2-t)! *n^(n-2-t), t=1..n-2) end: T:= proc(n, m) if m=1 then cy(n) else add (binomial(n-1, j) *cy(j+1) * T(n-1-j, m-1), j=2..n-1) fi end: seq (seq (T(n, m), m=1..n), n=1..11); # Alois P. Heinz, Sep 15 2008

MATHEMATICA

nn=12; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; Range[0, nn]!CoefficientList[ Series[Exp[y(Log[1/(1-t)]/2-t/2-t^2/4)], {x, 0, nn}], {x, y}] //Grid  (* Geoffrey Critzer, Nov 04 2012 *)

CROSSREFS

Cf. A057500 and A106238 (similar formulae that can be used in the unlabeled case).

Sequence in context: A123652 A127622 A185294 * A202857 A055965 A067154

Adjacent sequences:  A106236 A106237 A106238 * A106240 A106241 A106242

KEYWORD

nonn,tabl

AUTHOR

Washington Bomfim, May 03 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified November 26 01:49 EST 2014. Contains 250017 sequences.