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A258427 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.

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