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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. 3
1, 12, 18, 112, 420, 336, 956, 6816, 12936, 7200, 7830, 95579, 324540, 414450, 178200, 62744, 1244466, 6755720, 14886300, 14355000, 5045040, 496518, 15537456, 127063596, 430572780, 699460740, 542341800, 161441280 (list; table; graph; refs; listen; history; text; internal format)
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

FORMULA

T(n,k) = A256061(n,k) - A255982(n,k).

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-Fran├žois Alcover, Feb 22 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A256061, A255982.

Sequence in context: A232384 A086301 A231094 * A166627 A018957 A238922

Adjacent sequences:  A258424 A258425 A258426 * A258428 A258429 A258430

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, May 29 2015

STATUS

approved

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Last modified November 13 11:09 EST 2018. Contains 317133 sequences. (Running on oeis4.)