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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

COMMENTS

Also the Bell transform of A057500(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

LINKS

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

Washington Bomfim, Illustration of this sequence

FORMULA

E.g.f.: exp(-y/2*log(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_{i=1..n} A057500(i)^Ki and D = Product_{i=1..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

# The function BellMatrix is defined in A264428.

# Adds (1, 0, 0, 0, ..) as column 0.

a := n -> n!*n^(n-1)/2*add(1/(n^k*(n-k)!), k=3..n);

BellMatrix(n -> a(n+1), 9); # Peter Luschny, Jan 27 2016

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 * A271763 A271339 A202857

Adjacent sequences:  A106236 A106237 A106238 * A106240 A106241 A106242

KEYWORD

nonn,tabl

AUTHOR

Washington Bomfim, May 03 2005

STATUS

approved

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Last modified May 1 13:35 EDT 2016. Contains 272254 sequences.