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A036657 Number of n-node rooted unlabeled trees with out-degree <=2 and exactly 2 edges at the root.

%I A036657 #18 Jul 08 2025 21:53:20
%S A036657 0,0,0,1,1,3,5,12,23,52,109,244,532,1196,2671,6055,13726,31380,71901,
%T A036657 165635,382610,887215,2062777,4810230,11243898,26346341,61863991,
%U A036657 145560102,343121676,810246372,1916417479,4539722153,10769366928,25582243781,60847034869
%N A036657 Number of n-node rooted unlabeled trees with out-degree <=2 and exactly 2 edges at the root.
%H A036657 Alois P. Heinz, <a href="/A036657/b036657.txt">Table of n, a(n) for n = 0..2500</a>
%H A036657 S. J. Cyvin, J. Brunvoll, B. N. Cyvin, <a href="https://doi.org/10.1016/0166-1280(95)04329-6">Enumeration of constitutional isomers of polyenes</a>, J. Molec. Struct. (Theochem) 357, no. 3 (1995) 255-261.
%H A036657 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A036657 G.f. satisfies A(x) = (W(x)-x)*(1-x)/x-x, where W(x) is g.f. for A001190.
%p A036657 N := 40: G036657 := series(G001190*(1/x-1)-1,x,N); A036657 := n->coeff(G036657,x,n);
%t A036657 terms = 36; A[_] = 0;
%t A036657 Do[A[x_] = x + (1/2) (A[x]^2 + A[x^2]) + O[x]^terms // Normal, {terms}];
%t A036657 CoefficientList[1 + A[x], x] // Differences (* _Jean-François Alcover_, Oct 24 2019 *)
%Y A036657 First differences of A001190.
%K A036657 nonn
%O A036657 0,6
%A A036657 _N. J. A. Sloane_