%I #18 Feb 12 2023 10:23:31
%S 1,1,1,1,2,3,6,11,23,46,104,230,539,1270,3081,7536,18785,47207,120074,
%T 307739,795426,2069248,5418014,14263084,37742929,100331646,267854040,
%U 717863832,1930888297,5210968114,14106844554
%N Number of trees on n unlabeled nodes with all nodes of degree <= 7.
%H R. Otter, <a href="http://www.jstor.org/stable/1969046">The number of trees</a>, Ann. of Math. (2) 49 (1948), 583-599 discusses asymptotics.
%H E. M. Rains and N. J. A. Sloane, <a href="https://cs.uwaterloo.ca/journals/JIS/cayley.html">On Cayley's Enumeration of Alkanes (or 4-Valent Trees)</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F 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
%t n = 30; (* algorithm from Rains and Sloane *)
%t m = 7; (* maximum degree of node *)
%t CIm[f_, h_, x_] = SymmetricGroupIndex[m-1, x] /. x[i_] -> f[h, x^i];
%t CI[f_, h_, x_] = SymmetricGroupIndex[m, x] /. x[i_] -> f[h, x^i];
%t T[-1, z_] := 1; T[h_, z_] := T[h, z] = Table[z^k, {k, 0, n}] .
%t Take[CoefficientList[z^(n+1) + 1 + CIm[T, h-1, z] z, z], n+1];
%t 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 - T[h-1, z])^2/2 + (T[h, z^2] - T[h-1, z^2])/2, z], n+1], {h, 0, n/2}],
%t 1->1] (* end of original program *)
%t b[n_, i_, t_, k_] := b[n,i,t,k] = If[i<1, 0, Sum[Binomial[b[i-1,i-1,
%t k,k] + j-1, j]* b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]];
%t b[0, i_, t_, k_] = 1; m = 6; (* m = maximum children *) n = 40;
%t gf[x_] = 1 + Sum[b[j-1,j-1,m,m]x^j,{j,1,n}]; (* G.f. for A036722 *)
%t ci[x_] = SymmetricGroupIndex[m+1, x] /. x[i_] -> gf[x^i];
%t CoefficientList[Normal[Series[gf[x] - (gf[x]^2 - gf[x^2])/2 + x ci[x],
%t {x, 0, n}]],x] (* _Robert A. Russell_, Jan 19 2023 *)
%Y Column k=7 of A144528; A036722 (rooted trees).
%K nonn
%O 0,5
%A _Robert A. Russell_, Dec 29 2022
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