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A052250
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Triangle T(n,k) (n >= 1, k >= 1) giving dimension of bigrading of Hopf algebra of rooted trees.
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7
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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LINKS
| Alois P. Heinz, Rows n = 1..141, flattened
D. J. Broadhurst and D. Kreimer, Towards cohomology of renormalization...
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EXAMPLE
| 1;
1, 1;
1, 2, 1;
2, 3, 3, 1;
3, 6, 6, 4, 1;
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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}]] (* From Jean-François Alcover, Jan 20 2012, translated from Alois P. Heinz's Maple program *)
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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 * A099569 A191579 A097724
Adjacent sequences: A052247 A052248 A052249 * A052251 A052252 A052253
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KEYWORD
| nonn,nice,tabl
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AUTHOR
| David Broadhurst (D.Broadhurst(AT)open.ac.uk), Feb 05 2000
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EXTENSIONS
| More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 23 2009
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