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A238415 Triangle read by rows: T(n,k) is the number of trees with n vertices having k branching vertices (n>=2, 0<=k<=floor(n/2) - 1).

%I A238415 #14 May 23 2018 16:43:43
%S A238415 1,1,1,1,1,2,1,4,1,1,7,3,1,11,10,1,1,17,24,5,1,25,56,22,2,1,36,114,74,
%T A238415 10,1,50,224,219,55,2,1,70,411,576,224,19,1,94,733,1394,807,126,4,1,
%U A238415 127,1252,3150,2536,637,38,1,168,2091,6733,7305,2711,305,6
%N A238415 Triangle read by rows: T(n,k) is the number of trees with n vertices having k branching vertices (n>=2, 0<=k<=floor(n/2) - 1).
%C A238415 A branching node of a tree is a vertex of degree at least 3.
%C A238415 Sum of entries in row n is A000055(n) (number of trees with n vertices).
%C A238415 Row n has floor(n/2) entries.
%C A238415 T(n,1) = A004250(n-1).
%H A238415 Andrew Howroyd, <a href="/A238415/b238415.txt">Table of n, a(n) for n = 2..1226</a>
%F A238415 The author knows of no formula for T(n,k). The entries have been obtained in the following manner, explained for row n = 7. In A235111 we find that the 11 (= A000055(7)) trees with 7 vertices have M-indices 25, 27, 30, 35, 36, 40, 42, 48, 49, 56, and 64 (the M-index of a tree t is the smallest of the Matula numbers of the rooted trees isomorphic, as a tree, to t). Making use of the formula in A196049, from these Matula numbers one obtains that these trees have 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, and 1 branching vertices, respectively; the frequencies of 0, 1, and 2 are 1, 7, and 3, respectively. See the Maple program.
%e A238415 Row n=4 is T(4,0)=1,T(4,1)=1; indeed, the path P[4] has no branching vertex and the star S[4] has 1 branching vertex.
%e A238415 Triangle starts:
%e A238415 1;
%e A238415 1;
%e A238415 1, 1;
%e A238415 1, 2;
%e A238415 1, 4, 1;
%e A238415 1, 7, 3;
%e A238415 1, 11, 10, 1;
%e A238415 1, 17, 24, 5;
%p A238415 MI := [25, 27, 30, 35, 36, 40, 42, 48, 49, 56, 64]: with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 and bigomega(pi(n)) <> 2 then a(pi(n)) elif bigomega(n) = 1 then a(pi(n))+1 elif bigomega(s(n)) <> 2 then a(r(n))+a(s(n)) else a(r(n))+a(s(n))+1 end if end proc: g := add(x^a(MI[j]), j = 1 .. nops(MI)): seq(coeff(g, x, q), q = 0 .. 2);
%o A238415 (PARI)
%o A238415 EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
%o A238415 R(n)={my(v=[1]); for(i=2, n, v=concat([1], y*EulerMT(v) + (1-y)*v)); v}
%o A238415 seq(n)={my(p=x*Ser(R(n))); Vec(p + (((1-y)*x-1)*p^2 + ((1-y)*x+1)*substvec(p,[x,y],[x^2,y^2]))/2)}
%o A238415 { my(A=Vec(seq(20))); for(n=2, #A, print(Vecrev(A[n]))) } \\ _Andrew Howroyd_, May 21 2018
%Y A238415 Cf. A000055, A235111, A196049, A004250.
%K A238415 nonn,tabf
%O A238415 2,6
%A A238415 _Emeric Deutsch_, Mar 05 2014
%E A238415 Terms a(51) and beyond from _Andrew Howroyd_, May 21 2018