A052250 Triangle T(n,k) (n >= 1, k >= 1) giving dimension of bigrading of Hopf algebra of rooted trees.

1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 3, 6, 6, 4, 1, 8, 11, 13, 10, 5, 1, 16, 26, 27, 24, 15, 6, 1, 41, 58, 63, 55, 40, 21, 7, 1, 98, 142, 148, 132, 100, 62, 28, 8, 1, 250, 351, 363, 322, 251, 168, 91, 36, 9, 1, 631, 890, 912, 804, 635, 444, 266, 128, 45, 10, 1, 1646, 2282, 2330, 2051

Offset: 1

Author

David Broadhurst, Feb 05 2000

Keywords

• nonn
• nice
• tabl

Examples

			Triangle begins
  1;
  1, 1;
  1, 2, 1;
  2, 3, 3, 1;
  3, 6, 6, 4, 1;
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• D. J. Broadhurst and D. Kreimer, Towards cohomology of renormalization...

Crossrefs

First few columns give A051573, A051603, A052251, A052252.

Row sums give A000081(n+1). - Alois P. Heinz, Oct 23 2009

Programs

Maple

with(numtheory): A81:= proc(n) option remember; `if`(n<2, n, (add(add(d*A81(d), d=divisors(j)) *A81(n-j), j=1..n-1))/ (n-1)) end: b:= proc(n) option remember; -`if`(n<0, 1, add(b(n-i) *A81(i+1), i=1..n+1)) end: B:= proc(n) add(b(i) *x^i, i=0..n) end: T:= (n,k)-> coeff(B(n)^k, x, n-k): seq(seq(T(n, k), k=1..n), n=1..13); # Alois P. Heinz, Oct 23 2009

Mathematica

A81[n_] := A81[n] = If[n < 2, n, Sum[ Sum[ d*A81[d], {d, Divisors[j]} ] * A81[n-j], {j, 1, n-1}]/(n-1)]; b[n_] := b[n] = -If[n < 0, 1, Sum[ b[n-i]*A81[i+1], {i, 1, n+1}]]; B[n_] := Sum[ b[i]*x^i, {i, 0, n}]; T[n_, k_] := Coefficient[ B[n]^k, x, n-k]; Flatten[ Table[ T[n, k], {n, 1, 12}, {k, 1, n}]] (* Jean-François Alcover, Jan 20 2012, translated from Alois P. Heinz's Maple program *)

Extensions

More terms from Alois P. Heinz, Oct 23 2009