OFFSET
3,2
COMMENTS
T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. T(n,k) = 0 for k in {0, 1} or k>=n.
LINKS
Alois P. Heinz, Rows n = 3..135, flattened
EXAMPLE
T(3,2) = 1. There are A256061(3,2) = 30 binary trees with 3 inner nodes of exactly 2 different dimensions, 28 of them have unique hypercube partitions, 2 of them have the same partition:
: : : partition :
|--------------|---------------------|-----------|
| | (1) [2] | |
| | / \ / \ | .___. |
| trees: | [2] [2] (1) (1) | |_|_| |
| | / \ / \ / \ / \ | |_|_| |
| balanced | | |
| parentheses: | ([])[] [()]() | |
|--------------|---------------------|-----------|
Triangle T(n,k) begins:
.
. .
. . .
. . 1, .
. . 12, 18, .
. . 112, 420, 336, .
. . 956, 6816, 12936, 7200, .
. . 7830, 95579, 324540, 414450, 178200, .
. . 62744, 1244466, 6755720, 14886300, 14355000, 5045040, .
MAPLE
A:= proc(n, k) option remember; k^n*binomial(2*n, n)/(n+1) end:
B:= proc(n, k) option remember;
add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
end:
b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
H(n-1, k), add(H(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
end:
H:= proc(n, k) option remember; `if`(n=0, 1,
-add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
end:
G:= proc(n, k) option remember;
add(H(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
end:
T:= (n, k)-> B(n, k)-G(n, k):
seq(seq(T(n, k), k=2..n-1), n=3..12);
MATHEMATICA
A[n_, k_] := A[n, k] = k^n*Binomial[2*n, n]/(n+1); B[n_, k_] := B[n, k] = Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; b[n_, k_, t_] := b[n, k, t] = If[t==0, 1, If[t==1, H[n-1, k], Sum[H[j, k]*b[n-j-1, k, t-1], {j, 0, n-2}]]]; H[n_, k_] := H[n, k] = If[n==0, 1, -Sum[Binomial[k, j]* (-1)^j* b[n+1, k, 2^j], {j, 1, k}]]; G[n_, k_] := G[n, k] = Sum[H[n, k-i]*(-1)^i* Binomial[k, i], {i, 0, k}]; T[n_, k_] := T[n, k] = B[n, k]-G[n, k]; Table[Table[T[n, k], {k, 2, n-1}], {n, 3, 12}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 29 2015
STATUS
approved