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
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
Programs
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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
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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