A038096
Number of rooted graphs on n labeled nodes where the root has degree 3.
Original entry on oeis.org
32, 1280, 61440, 4587520, 587202560, 135291469824, 57724360458240, 46443371157258240, 71337018097548656640, 211030752203237270487040, 1210134745434243803880882176, 13518305228996352601898436526080
Offset: 4
For n=4, take 4 nodes labeled a,b,c,d. We can choose the root in 4 ways, say a, and it must be joined to b,c,d. Each of the three edges bc, bd, cd may or may not exist, so there are 4*8 = 32 = a(4) possibilities.
-
Table[n Binomial[n-1,3] 2^Binomial[n-1,2],{n,4,20}] (* Harvey P. Dale, Sep 14 2011 *)
-
a(n) = {n*binomial(n-1,3)*2^binomial(n-1,2)} \\ Andrew Howroyd, Nov 23 2020
A103904
a(n) = n*(n-1)/2 * 2^(n*(n-1)/2).
Original entry on oeis.org
0, 2, 24, 384, 10240, 491520, 44040192, 7516192768, 2473901162496, 1583296743997440, 1981583836043018240, 4869940435459321626624, 23574053482485268906770432, 225305087149939210031640608768
Offset: 1
- M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97, doi:10.1006/jcta.1996.2725.
- N. Elkies, G. Kuperberg, M. Larsen and J. Propp, Alternating sign matrices and domino tilings, Journal of Algebraic Combinatorics 1 (1992), 111-132 (Part I), 219-234 (Part II); arXiv:math/9201305 [math.CO], 1992.
- H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
- C. Krattenthaler, Schur function identities and the number of perfect matchings of Aztec holey rectangles, arXiv:math/9712204 [math.CO], 1997.
- Mathematics Stack Exchange, Mistake in OEIS A103904?, 2021.
Name replaced by a formula, a(1) changed from 1 to 0, and entry edited by
Andrey Zabolotskiy, Jun 05 2022
A038095
Number of rooted connected graphs on n labeled nodes where the root has degree 2.
Original entry on oeis.org
6, 72, 1440, 49680, 2998800, 324237312, 64440883584, 24059497893120, 17143668999705600, 23569875858252303360, 63000019679242001900544, 329150325651711743768150016, 3374625529825460904919664793600, 68094821953233373962606732799672320
Offset: 3
-
seq(n)={Vec(serlaplace(sum(k=1, n, k*binomial(k-1, 2)*2^binomial(k-1, 2)*x^k/k!)/sum(k=0, n, 2^binomial(k, 2)*x^k/k!) + O(x*x^n)))} \\ Andrew Howroyd, Nov 23 2020
A038097
Number of rooted connected graphs on n labeled nodes where the root has degree 3.
Original entry on oeis.org
32, 1120, 53760, 4155200, 550305280, 129990260736, 56369709634560, 45808126727193600, 70779622448719134720, 210103333009795315650560, 1207180278201294640467288064, 13500153139563947729371140096000, 295095590701444457972767937903329280
Offset: 4
For n=4, take 4 nodes labeled a,b,c,d. We can choose the root in 4 ways, say a, and it must be joined to b,c,d. Each of the three edges bc, bd, cd may or may not exist, so there are 4*8 = 32 = a(4) possibilities.
-
seq(n)={Vec(serlaplace(sum(k=1, n, k*binomial(k-1,3)*2^binomial(k-1,2)*x^k/k!)/sum(k=0, n, 2^binomial(k, 2)*x^k/k!) + O(x*x^n)))} \\ Andrew Howroyd, Nov 23 2020
A285529
Triangle read by rows: T(n,k) is the number of nodes of degree k counted over all simple labeled graphs on n nodes, n>=1, 0<=k<=n-1.
Original entry on oeis.org
1, 2, 2, 6, 12, 6, 32, 96, 96, 32, 320, 1280, 1920, 1280, 320, 6144, 30720, 61440, 61440, 30720, 6144, 229376, 1376256, 3440640, 4587520, 3440640, 1376256, 229376, 16777216, 117440512, 352321536, 587202560, 587202560, 352321536, 117440512, 16777216
Offset: 1
1,
2, 2,
6, 12, 6,
32, 96, 96, 32,
320, 1280, 1920, 1280, 320,
...
-
nn = 9; Map[Select[#, # > 0 &] &,
Drop[Transpose[Table[A[z_] := Sum[Binomial[n, k] 2^Binomial[n, 2] z^n/n!, {n, 0, nn}];Range[0, nn]! CoefficientList[Series[z A[z], {z, 0, nn}], z], {k,0, nn - 1}]], 1]] // Grid
Showing 1-5 of 5 results.
Comments