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%I #24 Jan 22 2021 20:26:44
%S 1,1,1,2,6,4,5,30,51,26,12,146,474,576,236,33,719,3950,8572,8060,2752,
%T 90,3590,31464,108416,175380,134136,39208,261,18283,245916,1262732,
%U 3124650,4014348,2584568,660032,766,94648,1908858,14047288,49885320,95715728,101799712,56555904,12818912
%N Triangle read by rows: T(n,k) is the number of lone-child-avoiding rooted trees with n leaves of exactly k colors.
%C Lone-child-avoiding rooted trees are also called planted series-reduced trees in some other sequences.
%H Andrew Howroyd, <a href="/A319376/b319376.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%F T(n,k) = Sum_{i=1..k} (-1)^(k-i)*binomial(k,i)*A319254(n,i).
%F Sum_{k=1..n} k * T(n,k) = A326396(n). - _Alois P. Heinz_, Sep 11 2019
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 6, 4;
%e 5, 30, 51, 26;
%e 12, 146, 474, 576, 236;
%e 33, 719, 3950, 8572, 8060, 2752;
%e 90, 3590, 31464, 108416, 175380, 134136, 39208;
%e 261, 18283, 245916, 1262732, 3124650, 4014348, 2584568, 660032;
%e ...
%e From _Gus Wiseman_, Dec 31 2020: (Start)
%e The 12 trees counted by row n = 3:
%e (111) (112) (123)
%e (1(11)) (122) (1(23))
%e (1(12)) (2(13))
%e (1(22)) (3(12))
%e (2(11))
%e (2(12))
%e (End)
%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
%p end:
%p A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):
%p T:= (n, k)-> add(A(n, k-j)*(-1)^j*binomial(k, j), j=0..k-1):
%p seq(seq(T(n, k), k=1..n), n=1..10); # _Alois P. Heinz_, Sep 18 2018
%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A[i, k] + j - 1, j] b[n - i j, i - 1, k], {j, 0, n/i}]]];
%t A[n_, k_] := If[n < 2, n k, b[n, n - 1, k]];
%t T[n_, k_] := Sum[(-1)^(k - i)*Binomial[k, i]*A[n, i], {i, 1, k}];
%t Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 24 2019, after _Alois P. Heinz_ *)
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t mtot[m_]:=Prepend[Join@@Table[Tuples[mtot/@p],{p,Select[mps[m],1<Length[#]<Length[m]&]}],m];
%t allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t Table[Sum[Length[Union[mtot[s]]],{s,Select[allnorm[n],Length[Union[#]]==k&]}],{n,0,5},{k,0,n}] (* _Gus Wiseman_, Dec 31 2020 *)
%o (PARI) \\ here R(n,k) is k-th column of A319254.
%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o R(n, k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v, [0]))[n])); v}
%o M(n)={my(v=vector(n, k, R(n,k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k,i)*v[i])))}
%o {my(T=M(10)); for(n=1, #T~, print(T[n, ][1..n]))}
%Y Columns k=1..2 are A000669, A319377.
%Y Main diagonal is A000311.
%Y Row sums are A316651.
%Y Cf. A141610, A242249, A255517, A256064, A256068, A319254, A319541, A326396.
%Y The unlabeled version, counting inequivalent leaf-colorings of lone-child-avoiding rooted trees, is A330465.
%Y Lone-child-avoiding rooted trees are counted by A001678 (shifted left once).
%Y Labeled lone-child-avoiding rooted trees are counted by A060356.
%Y Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
%Y Cf. A000014, A000081, A000169, A005804, A206429, A330951.
%K nonn,tabl
%O 1,4
%A _Andrew Howroyd_, Sep 17 2018
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