A055544
Total number of nodes in all rooted trees with n nodes.
Original entry on oeis.org
1, 2, 6, 16, 45, 120, 336, 920, 2574, 7190, 20262, 57192, 162318, 461622, 1317165, 3766096, 10792399, 30980862, 89084844, 256524560, 739658472, 2135213982, 6170505665, 17849399616, 51679366125, 149750549260, 434260826883, 1260197855532, 3659410074932
Offset: 1
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with(numtheory):
b:= proc(n) option remember; local d, j; `if`(n<2, n,
(add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/ (n-1))
end:
a:= n-> n*b(n):
seq(a(n), n=1..30); # Alois P. Heinz, May 16 2013
-
b[n_] := b[n] = If[n<2, n, (Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}])/(n-1)]; a[n_] := n*b[n]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jan 25 2017, after Alois P. Heinz *)
A055542
Total number of nodes in all simple graphs of n nodes.
Original entry on oeis.org
1, 4, 12, 44, 170, 936, 7308, 98768, 2472012, 120051680, 11208976504, 1981094071104, 656526407783376, 406758179201296832, 471397289547064631520, 1024016251272440926318848, 4180909690610059855623236192, 32176399052621010609861807435264
Offset: 1
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b:= proc(n, i, l) `if`(n=0 or i=1, 1/n!*2^((p-> add(ceil((p[j]-1)/2)
+add(igcd(p[k], p[j]), k=1..j-1), j=1..nops(p)))([l[], 1$n])),
add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i))
end:
a:= n-> n*b(n$2, []):
seq(a(n), n=1..20); # Alois P. Heinz, Aug 14 2019
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Needs["Combinatorica`"];
Table[NumberOfGraphs[n]*n,{n,1,20}] (* Geoffrey Critzer, Oct 13 2012 *)
A095349
Total number of edges in all trees on n nodes.
Original entry on oeis.org
0, 1, 2, 6, 12, 30, 66, 161, 376, 954, 2350, 6061, 15612, 41067, 108374, 289800, 778064, 2105739, 5723190, 15638235, 42890100, 118098876, 326217628, 903897631, 2511285360, 6994836250, 19527701960, 54632961864, 153147864380, 430095282258, 1209924870900
Offset: 1
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nn = 25; f[x_] := Sum[a[n] x^n, {n, 0, nn}]; sol = SolveAlways[0 == Series[
f[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}], x];
r[x_] := Sum[a[n] x^n, {n, 0, nn}] /. sol; Drop[Level[CoefficientList[
Series[x^2 D[1/x (r[x] - 1/2 (r[x]^2 - r[x^2])), x], {x, 0, nn}],
x], {2}], 1] (* Geoffrey Critzer, Jul 06 2020 *)
A345133
Decimal expansion of the limit, as n approaches infinity, of the probability that a node is a leaf in a free tree with n nodes.
Original entry on oeis.org
4, 3, 8, 1, 5, 6, 2, 3, 5, 6, 6, 4
Offset: 0
- Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.6.3, p. 304.
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