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

%I A052250 #25 May 25 2019 01:28:13
%S A052250 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,
%T A052250 58,63,55,40,21,7,1,98,142,148,132,100,62,28,8,1,250,351,363,322,251,
%U A052250 168,91,36,9,1,631,890,912,804,635,444,266,128,45,10,1,1646,2282,2330,2051
%N A052250 Triangle T(n,k) (n >= 1, k >= 1) giving dimension of bigrading of Hopf algebra of rooted trees.
%H A052250 Alois P. Heinz, <a href="/A052250/b052250.txt">Rows n = 1..141, flattened</a>
%H A052250 D. J. Broadhurst and D. Kreimer, <a href="http://arXiv.org/abs/hep-th/0001202">Towards cohomology of renormalization...</a>
%e A052250 Triangle begins
%e A052250   1;
%e A052250   1, 1;
%e A052250   1, 2, 1;
%e A052250   2, 3, 3, 1;
%e A052250   3, 6, 6, 4, 1;
%p A052250 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
%t A052250 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 *)
%Y A052250 First few columns give A051573, A051603, A052251, A052252.
%Y A052250 Row sums give A000081(n+1). - _Alois P. Heinz_, Oct 23 2009
%K A052250 nonn,nice,tabl
%O A052250 1,5
%A A052250 _David Broadhurst_, Feb 05 2000
%E A052250 More terms from _Alois P. Heinz_, Oct 23 2009