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%I #12 Sep 07 2018 16:46:50
%S 1,12,18,112,420,336,956,6816,12936,7200,7830,95579,324540,414450,
%T 178200,62744,1244466,6755720,14886300,14355000,5045040,496518,
%U 15537456,127063596,430572780,699460740,542341800,161441280
%N Number T(n,k) of redundant binary trees with n inner nodes of exactly k different dimensions used for the partition of the k-dimensional hypercube by hierarchical bisection; triangle T(n,k), n>=3, 2<=k<=n-1, read by rows.
%C 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.
%H Alois P. Heinz, <a href="/A258427/b258427.txt">Rows n = 3..135, flattened</a>
%F T(n,k) = A256061(n,k) - A255982(n,k).
%e 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:
%e : : : partition :
%e |--------------|---------------------|-----------|
%e | | (1) [2] | |
%e | | / \ / \ | .___. |
%e | trees: | [2] [2] (1) (1) | |_|_| |
%e | | / \ / \ / \ / \ | |_|_| |
%e | balanced | | |
%e | parentheses: | ([])[] [()]() | |
%e |--------------|---------------------|-----------|
%e Triangle T(n,k) begins:
%e .
%e . .
%e . . .
%e . . 1, .
%e . . 12, 18, .
%e . . 112, 420, 336, .
%e . . 956, 6816, 12936, 7200, .
%e . . 7830, 95579, 324540, 414450, 178200, .
%e . . 62744, 1244466, 6755720, 14886300, 14355000, 5045040, .
%p A:= proc(n, k) option remember; k^n*binomial(2*n, n)/(n+1) end:
%p B:= proc(n, k) option remember;
%p add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
%p end:
%p b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
%p H(n-1, k), add(H(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
%p end:
%p H:= proc(n, k) option remember; `if`(n=0, 1,
%p -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
%p end:
%p G:= proc(n, k) option remember;
%p add(H(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
%p end:
%p T:= (n, k)-> B(n, k)-G(n, k):
%p seq(seq(T(n, k), k=2..n-1), n=3..12);
%t 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_ *)
%Y Cf. A256061, A255982.
%K nonn,tabl
%O 3,2
%A _Alois P. Heinz_, May 29 2015
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