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

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, 7, 33, 99, 192, 231, 157, 46, 1, 8, 44, 157, 382, 615, 634, 373, 98, 1, 9, 56, 234, 682, 1380, 1905, 1704, 890, 207, 1, 10, 70, 333, 1133, 2755, 4782, 5746, 4554, 2130, 451

Offset: 2

Author

N. J. A. Sloane, May 29 2005

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  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, 7, 33,  99, 192, 231, 157,  46;
  1, 8, 44, 157, 382, 615, 634, 373, 98;
  ...
		

References

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

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..1276

Crossrefs

Right diagonal is A001190.

Row sums give A000669.

Cf. A001678.

Programs

PARI

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)}
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])}
{ my(T=A(10)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Sep 01 2018

Formula

A001678(n) = Sum_{i=2..n-2} T(i, n-1-i) for n >= 3. - Marko Riedel, Mar 29 2021

Extensions

Name clarified and terms a(38) and beyond from Andrew Howroyd, Sep 01 2018