login
Triangle read by rows: T(n,k) is the number of series-reduced planted trees with n leaves and k internal nodes.
2

%I #16 Mar 30 2021 09:38:30

%S 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 7,33,99,192,231,157,46,1,8,44,157,382,615,634,373,98,1,9,56,234,682,

%U 1380,1905,1704,890,207,1,10,70,333,1133,2755,4782,5746,4554,2130,451

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

%D J. Riordan, The blossoming of Schroeder's fourth problem, Acta Math., 137 (1976), 1-16.

%H Andrew Howroyd, <a href="/A106179/b106179.txt">Table of n, a(n) for n = 2..1276</a>

%F A001678(n) = Sum_{i=2..n-2} T(i, n-1-i) for n >= 3. - _Marko Riedel_, Mar 29 2021

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 2, 2;

%e 1, 3, 5, 3;

%e 1, 4, 10, 12, 6;

%e 1, 5, 16, 29, 28, 11;

%e 1, 6, 24, 57, 84, 66, 23;

%e 1, 7, 33, 99, 192, 231, 157, 46;

%e 1, 8, 44, 157, 382, 615, 634, 373, 98;

%e ...

%o (PARI)

%o 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 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 { my(T=A(10)); for(n=1, #T, print(T[n])) } \\ _Andrew Howroyd_, Sep 01 2018

%Y Right diagonal is A001190.

%Y Row sums give A000669.

%Y Cf. A001678.

%K nonn,tabl

%O 2,5

%A _N. J. A. Sloane_, May 29 2005

%E Name clarified and terms a(38) and beyond from _Andrew Howroyd_, Sep 01 2018