A055543
Total number of nodes in all trees with n nodes.
Original entry on oeis.org
1, 2, 3, 8, 15, 36, 77, 184, 423, 1060, 2585, 6612, 16913, 44226, 116115, 309120, 826693, 2229606, 6041145, 16461300, 45034605, 123722632, 341045702, 943197528, 2615922250, 7274629700, 20278767420, 56656404896, 158617430965, 444926154060, 1250255699930
Offset: 1
-
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; CoefficientList[Series[x D[r[x] - 1/2 (r[x]^2 - r[x^2]), x], {x, 0, nn}], x] (* Geoffrey Critzer, Jul 06 2020 *)
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
-
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
-
Needs["Combinatorica`"];
Table[NumberOfGraphs[n]*n,{n,1,20}] (* Geoffrey Critzer, Oct 13 2012 *)
A095350
Total number of edges in all rooted trees on n nodes.
Original entry on oeis.org
0, 1, 4, 12, 36, 100, 288, 805, 2288, 6471, 18420, 52426, 149832, 428649, 1229354, 3530715, 10157552, 29259703, 84396168, 243698332, 704436640, 2038158801, 5902222810, 17105674632, 49612191480, 143990912750, 418177092554
Offset: 1
A275331
Triangle read by rows, T(n,k) = k*Sum_{m=1..n/k} t(k)*t(n-k*m+1) with t = A000081, for n>=1 and 1<=k<=n.
Original entry on oeis.org
1, 2, 2, 4, 2, 6, 8, 6, 6, 16, 17, 10, 12, 16, 45, 37, 24, 30, 32, 45, 120, 85, 50, 60, 64, 90, 120, 336, 200, 120, 132, 160, 180, 240, 336, 920, 486, 280, 318, 336, 405, 480, 672, 920, 2574, 1205, 692, 750, 800, 945, 1080, 1344, 1840, 2574, 7190
Offset: 1
Triangle starts:
[n] [k=1,2,...] row sum
[1] [1] 1
[2] [2, 2] 4
[3] [4, 2, 6] 12
[4] [8, 6, 6, 16] 36
[5] [17, 10, 12, 16, 45] 100
[6] [37, 24, 30, 32, 45, 120] 288
[7] [85, 50, 60, 64, 90, 120, 336] 805
[8] [200, 120, 132, 160, 180, 240, 336, 920] 2288
[9] [486, 280, 318, 336, 405, 480, 672, 920, 2574] 6471
A308433
G.f.: x * (d/dx) x * Product_{k>=1} (1 + x^k)^(a(k)/k).
Original entry on oeis.org
1, 2, 3, 8, 15, 36, 84, 200, 468, 1130, 2717, 6576, 15938, 38780, 94485, 230816, 564553, 1383318, 3393742, 8336960, 20502216, 50472928, 124369832, 306729456, 757078000, 1870040822, 4622317812, 11432698704, 28294211920, 70063292310, 173584768088, 430276174016, 1067049650238
Offset: 1
-
a[n_] := a[n] = SeriesCoefficient[x D[x Product[(1 + x^k)^(a[k]/k), {k, 1, n - 1}], x], {x, 0, n}]; Table[a[n], {n, 1, 33}]
a[n_] := a[n] = n SeriesCoefficient[x Exp[Sum[Sum[(-1)^(k/d + 1) a[d], {d, Divisors[k]}] x^k/k, {k, 1, n - 1}]], {x, 0, n}]; Table[a[n], {n, 1, 33}]
a[n_] := a[n] = Sum[a[n - k] Sum[(-1)^(k/d + 1) d a[d], {d, Divisors[k]}], {k, 1, n - 1}]/(n - 1); a[1] = 1; Table[n a[n], {n, 1, 33}]
Showing 1-5 of 5 results.