OFFSET
1,5
COMMENTS
See table 2.2 in the Johnson reference.
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
Virginia Perkins Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. South Carolina, 2012.
FORMULA
T(n,k) = Sum_{i=1..k} (-1)^(k-i)*binomial(k,i)*A319539(n,i).
EXAMPLE
Triangle begins:
1;
1, 1;
1, 4, 3;
2, 14, 27, 15;
3, 48, 180, 240, 105;
6, 171, 1089, 2604, 2625, 945;
11, 614, 6333, 24180, 42075, 34020, 10395;
23, 2270, 36309, 207732, 554820, 755370, 509355, 135135;
...
MAPLE
A:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
(t-> t*(1-t)/2)(A(n/2, k)))+add(A(i, k)*A(n-i, k), i=1..n/2))
end:
T:= (n, k)-> add((-1)^i*binomial(k, i)*A(n, k-i), i=0..k):
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Sep 23 2018
MATHEMATICA
A[n_, k_] := A[n, k] = If[n<2, k n, If[OddQ[n], 0, (#(1-#)/2)&[A[n/2, k]]] + Sum[A[i, k] A[n - i, k], {i, 1, n/2}]];
T[n_, k_] := Sum[(-1)^i Binomial[k, i] A[n, k - i], {i, 0, k}];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 02 2019, after Alois P. Heinz *)
PROG
(PARI) \\ here R(n, k) is k-th column of A319539 as a vector.
R(n, k)={my(v=vector(n)); v[1]=k; for(n=2, n, v[n]=sum(j=1, (n-1)\2, v[j]*v[n-j]) + if(n%2, 0, binomial(v[n/2]+1, 2))); v}
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])))}
{my(T=M(10)); for(n=1, #T~, print(T[n, ][1..n]))}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Sep 22 2018
STATUS
approved