OFFSET
1,7
COMMENTS
Row sums are A005195.
For k > n/2, T(n,k) = T(n-1,k-1). - Geoffrey Critzer, Oct 13 2012
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Part 12 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 6 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Forest
FORMULA
T(n, k) = sum over the partitions of n, 1M1 + 2M2 + ... + nMn, with exactly k parts, of Product_{i=1..n} binomial(A000055(i) + Mi - 1, Mi). - Washington Bomfim, May 12 2005
EXAMPLE
Triangle begins:
1;
1, 1;
1, 1, 1;
2, 2, 1, 1;
3, 3, 2, 1, 1;
6, 6, 4, 2, 1, 1;
11, 11, 7, 4, 2, 1, 1;
23, 23, 14, 8, 4, 2, 1, 1;
47, 46, 29, 15, 8, 4, 2, 1, 1;
106, 99, 60, 32, 16, 8, 4, 2, 1, 1;
...
MAPLE
with(numtheory):
b:= proc(n) option remember; local d, j; `if` (n<=1, n,
(add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/(n-1))
end:
t:= proc(n) option remember; local k; `if` (n=0, 1,
b(n)-(add(b(k)*b(n-k), k=0..n)-`if`(irem(n, 2)=0, b(n/2), 0))/2)
end:
g:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
`if`(min(i, p)<1, 0, add(g(n-i*j, i-1, p-j) *
binomial(t(i)+j-1, j), j=0..min(n/i, p)))))
end:
a:= (n, k)-> g(n, n, k):
seq(seq(a(n, k), k=1..n), n=1..14); # Alois P. Heinz, Aug 20 2012
MATHEMATICA
nn=30; s[n_, k_]:=s[n, k]=a[n+1-k]+If[n<2k, 0, s[n-k, k]]; a[1]=1; a[n_]:=a[n]=Sum[a[i]s[n-1, i]i, {i, 1, n-1}]/(n-1); ft=Table[a[i]-Sum[a[j]a[i-j], {j, 1, i/2}]+If[OddQ[i], 0, a[i/2](a[i/2]+1)/2], {i, 1, nn}]; CoefficientList[Series[Product[1/(1-y x^i)^ft[[i]], {i, 1, nn}], {x, 0, 20}], {x, y}]//Grid (* Geoffrey Critzer, Oct 13 2012, after code given by Robert A. Russell in A000055 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Eric W. Weisstein, May 29 2004
EXTENSIONS
More terms from Vladeta Jovovic, Jun 03 2004
STATUS
approved