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%I #19 Dec 26 2020 11:09:30
%S 0,1,0,1,1,2,3,7,11,26,52,120,266,640,1509,3702,9090,22781,57452,
%T 146783,377357,978342,2550611,6690242,17633855,46705333,124227015,
%U 331757697,889207207,2391478247,6451880415,17457214729,47363110968
%N Number of centered 6-valent trees with n nodes.
%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 a(n) = A036653(n) - A036652(n).
%t n = 20; (* algorithm from Rains and Sloane *)
%t S5[f_,h_,x_] := f[h,x]^5/120 + f[h,x]^3 f[h,x^2]/12 + f[h,x]^2 f[h,x^3]/6 + f[h,x] f[h,x^2]^2/8 + f[h,x] f[h,x^4]/4 + f[h,x^2] f[h,x^3]/6 + f[h,x^5]/5;
%t S6[f_,h_,x_] := f[h,x]^6/720 + f[h,x]^4 f[h,x^2]/48 + f[h,x]^3 f[h,x^3]/18 + f[h,x]^2 f[h,x^2]^2/16 + f[h,x]^2 f[h,x^4]/8 + f[h,x] f[h,x^2] f[h,x^3]/6 + f[h,x] f[h,x^5]/5 + f[h,x^2]^3/48 + f[h,x^2] f[h,x^4]/8 + f[h,x^3]^2/18 + f[h,x^6]/6;
%t T[-1,z_] := 1; T[h_,z_] := T[h,z] = Table[z^k, {k,0,n}].Take[CoefficientList[z^(n+1) + 1 + S5[T,h-1,z]z, z], n+1];
%t Sum[Take[CoefficientList[z^(n+1) + S6[T,h-1,z]z - S6[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] (* _Robert A. Russell_, Sep 15 2018 *)
%Y Cf. A036652, A036653.
%K nonn
%O 0,6
%A _N. J. A. Sloane_
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