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A215862
Number of simple labeled graphs on n+2 nodes with exactly n connected components that are trees or cycles.
9
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
OFFSET
0,2
COMMENTS
Partial sums of A077414. - Bruno Berselli, Jul 30 2015
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.
Sequence in context: A186310 A122684 A122681 * A174612 A020496 A108484
KEYWORD
nonn,easy,changed
AUTHOR
Alois P. Heinz, Aug 25 2012
STATUS
approved