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A215862 Number of simple labeled graphs on n+2 nodes with exactly n connected components that are trees or cycles. 7
0, 4, 19, 55, 125, 245, 434, 714, 1110, 1650, 2365, 3289, 4459, 5915, 7700, 9860, 12444, 15504, 19095, 23275, 28105, 33649, 39974, 47150, 55250, 64350, 74529, 85869, 98455, 112375, 127720, 144584, 163064, 183260, 205275, 229215, 255189, 283309, 313690, 346450 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Partial sums of A077414. [Bruno Berselli, Jul 30 2015]

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

FORMULA

G.f.: (x-4)*x/(x-1)^5.

a(n) = C(n+2,3)*(3*n+13)/4.

a(n) = 5*a(n-1)- 10*a(n-2)+ 10*a(n-3) -5*a(n-4)+a(n-5), n>4. - Harvey P. Dale, Sep 10 2012

a(n) = 1/n! * Sum_{j=0..n} C(n,j)*(-1)^(n-j)*(j)^(n+1)*(j-1)). - Vladimir Kruchinin, Jun 06 2013

a(n) = 4*A000332(n+2) - A000332(n+1). - R. J. Mathar, Aug 12 2013

a(n) = Sum_{i=0..n} (3+i)*A000217(i). [Bruno Berselli, Apr 29 2014]

EXAMPLE

a(1) = 4:

.1-2.  .1-2.  .1-2.  .1 2.

.|/ .  .|. .  . / .  .|/ .

.3...  .3...  .3...  .3...

MAPLE

a:= n-> binomial(n+2, 3)*(3*n+13)/4:

seq(a(n), n=0..40);

MATHEMATICA

Table[Binomial[n+2, 3] (3n+13)/4, {n, 0, 40}] (* or *) LinearRecurrence[ {5, -10, 10, -5, 1}, {0, 4, 19, 55, 125}, 40] (* Harvey P. Dale, Sep 10 2012 *)

CROSSREFS

A diagonal of A215861.

Regarding the sixth formula, see similar sequences listed in A241765.

Cf. A000332, A077414.

Sequence in context: A186310 A122684 A122681 * A174612 A020496 A108484

Adjacent sequences:  A215859 A215860 A215861 * A215863 A215864 A215865

KEYWORD

nonn,easy

AUTHOR

Alois P. Heinz, Aug 25 2012

STATUS

approved

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Last modified August 18 08:57 EDT 2019. Contains 326077 sequences. (Running on oeis4.)