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%I #21 Dec 26 2020 11:08:17
%S 0,0,1,0,1,1,3,4,11,19,49,103,254,583,1445,3506,8815,22082,56286,
%T 143822,371354,963250,2516822,6607348,17440933,46233833,123090070,
%U 328923702,882114742,2373351473,6405275496,17336081498,47047112028
%N Number of bicentered 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) - A036651(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 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) + (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}] (* _Robert A. Russell_, Sep 15 2018 *)
%Y Cf. A036651, A036653.
%K nonn
%O 0,7
%A _N. J. A. Sloane_
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