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

2,5

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

Formula

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

Example

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;
  ...

Prog

(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

Crossrefs

Right diagonal is A001190.

Row sums give A000669.

Cf. A001678.

Sequence in context: A208336 A038137 A073133 * A081572 A292630 A144287

Adjacent sequences: A106176 A106177 A106178 * A106180 A106181 A106182

Keyword

nonn,tabl

Author

N. J. A. Sloane, May 29 2005

Extensions

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

Status

approved