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

1,5

Links

Alois P. Heinz, Rows n = 1..141, flattened

D. J. Broadhurst and D. Kreimer, Towards cohomology of renormalization...

Example

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

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 *)

Crossrefs

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

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

Sequence in context: A099567 A140530 A202191 * A333878 A099569 A191579

Adjacent sequences: A052247 A052248 A052249 * A052251 A052252 A052253

Keyword

nonn,nice,tabl

Author

David Broadhurst, Feb 05 2000

Extensions

More terms from Alois P. Heinz, Oct 23 2009

Status

approved