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A106179 Triangle read by rows: T(n,k) is the number of series-reduced planted trees with n leaves and k internal nodes.

%I A106179 #16 Mar 30 2021 09:38:30
%S A106179 1,1,1,1,2,2,1,3,5,3,1,4,10,12,6,1,5,16,29,28,11,1,6,24,57,84,66,23,1,
%T A106179 7,33,99,192,231,157,46,1,8,44,157,382,615,634,373,98,1,9,56,234,682,
%U A106179 1380,1905,1704,890,207,1,10,70,333,1133,2755,4782,5746,4554,2130,451
%N A106179 Triangle read by rows: T(n,k) is the number of series-reduced planted trees with n leaves and k internal nodes.
%D A106179 J. Riordan, The blossoming of Schroeder's fourth problem, Acta Math., 137 (1976), 1-16.
%H A106179 Andrew Howroyd, <a href="/A106179/b106179.txt">Table of n, a(n) for n = 2..1276</a>
%F A106179 A001678(n) = Sum_{i=2..n-2} T(i, n-1-i) for n >= 3. - _Marko Riedel_, Mar 29 2021
%e A106179 Triangle begins:
%e A106179   1;
%e A106179   1, 1;
%e A106179   1, 2,  2;
%e A106179   1, 3,  5,   3;
%e A106179   1, 4, 10,  12,   6;
%e A106179   1, 5, 16,  29,  28,  11;
%e A106179   1, 6, 24,  57,  84,  66,  23;
%e A106179   1, 7, 33,  99, 192, 231, 157,  46;
%e A106179   1, 8, 44, 157, 382, 615, 634, 373, 98;
%e A106179   ...
%o A106179 (PARI)
%o A106179 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 A106179 A(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n]=y*EulerMT(v[1..n])[n]); apply(p -> Vecrev(p/y), v[2..n])}
%o A106179 { my(T=A(10)); for(n=1, #T, print(T[n])) } \\ _Andrew Howroyd_, Sep 01 2018
%Y A106179 Right diagonal is A001190.
%Y A106179 Row sums give A000669.
%Y A106179 Cf. A001678.
%K A106179 nonn,tabl
%O A106179 2,5
%A A106179 _N. J. A. Sloane_, May 29 2005
%E A106179 Name clarified and terms a(38) and beyond from _Andrew Howroyd_, Sep 01 2018