A359392 Number of trees on n unlabeled nodes with all nodes of degree <= 7.

1, 1, 1, 1, 2, 3, 6, 11, 23, 46, 104, 230, 539, 1270, 3081, 7536, 18785, 47207, 120074, 307739, 795426, 2069248, 5418014, 14263084, 37742929, 100331646, 267854040, 717863832, 1930888297, 5210968114, 14106844554

Offset: 0

Author

Robert A. Russell, Dec 29 2022

Keywords

• nonn

Links

• R. Otter, The number of trees, Ann. of Math. (2) 49 (1948), 583-599 discusses asymptotics.
• E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
• Index entries for sequences related to trees

Crossrefs

Column k=7 of A144528; A036722 (rooted trees).

Programs

Mathematica

n = 30; (* algorithm from Rains and Sloane *)
m = 7; (* maximum degree of node *)
CIm[f_, h_, x_] =  SymmetricGroupIndex[m-1, x] /. x[i_] -> f[h, x^i];
CI[f_, h_, x_] = SymmetricGroupIndex[m, x] /. x[i_] -> f[h, x^i];
T[-1, z_] := 1; T[h_, z_] :=  T[h, z] = Table[z^k, {k, 0, n}] .
  Take[CoefficientList[z^(n+1) + 1 + CIm[T, h-1, z] z, z], n+1];
ReplacePart[Sum[Take[CoefficientList[z^(n+1) + CI[T, h-1, z] z - CI[T, h-2, z] z - (T[h-1, z] - T[h-2, z]) (T[h-1, z] - 1), z], n+1], {h, 1, n/2}] + PadRight[{0, 1}, n+1] + Sum[Take[CoefficientList[z^(n+1) + (T[h, z]
  - T[h-1, z])^2/2 + (T[h, z^2] - T[h-1, z^2])/2, z], n+1], {h, 0, n/2}],
  1->1] (* end of original program *)
b[n_, i_, t_, k_] := b[n,i,t,k] = If[i<1, 0, Sum[Binomial[b[i-1,i-1,
  k,k] + j-1, j]* b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]];
b[0, i_, t_, k_] = 1; m = 6; (* m = maximum children *) n = 40;
gf[x_] = 1 + Sum[b[j-1,j-1,m,m]x^j,{j,1,n}]; (* G.f. for A036722 *)
ci[x_] = SymmetricGroupIndex[m+1, x] /. x[i_] -> gf[x^i];
CoefficientList[Normal[Series[gf[x] - (gf[x]^2 - gf[x^2])/2 + x ci[x],
{x, 0, n}]],x] (* Robert A. Russell, Jan 19 2023 *)

Formula

G.f.: B(x) - cycle_index(S2,-B(x)) + x * cycle_index(S7,B(x)) = B(x) - (B(x)^2 - B(x^2)) / 2 + x * (B(x)^7 + 21 B(x)^5 B(x^2) + 105 B(x)^3 B(x^2)^2 + 105 B(x) B(x^2)^3 + 70 B(x)^4 B(x^3) + 420 B(x)^2 B(x^2) B(x^3) + 210 B(x^2)^2 B(x^3) + 280 B(x) B(x^3)^2 + 210 B(x)^3 B(x^4) + 630 B(x) B(x^2) B(x^4) + 420 B(x^3) B(x^4) + 504 B(x)^2 B(x^5) + 504 B(x^2) B(x^5) + 840 B(x) B(x^6) + 720 B(x^7)) / 5040, where B(x) = 1 + x * cycle_index(S6,B(x)) = 1 + x * (B(x)^6 + 15*B(x)^4*B(x^2) + 45*B(x)^2*B(x^2)^2 + 15*B(x^2)^3 + 40*B(x)^3*B(x^3) + 120*B(x)*B(x^2)*B(x^3) + 40*B(x^3)^2 + 90*B(x)^2*B(x^4) + 90*B(x^2)*B(x^4) + 144*B(x)*B(x^5) + 120*B(x^6)) / 720 is the generating function for A036722. - Robert A. Russell, Jan 19 2023