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A318758 Number T(n,k) of rooted trees with n nodes such that k equals the maximal number of isomorphic subtrees extending from the same node; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows. 12
1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 3, 4, 1, 1, 0, 6, 9, 3, 1, 1, 0, 12, 22, 9, 3, 1, 1, 0, 25, 54, 23, 8, 3, 1, 1, 0, 52, 138, 60, 23, 8, 3, 1, 1, 0, 113, 346, 164, 61, 22, 8, 3, 1, 1, 0, 247, 889, 443, 167, 61, 22, 8, 3, 1, 1, 0, 548, 2285, 1209, 461, 168, 60, 22, 8, 3, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

T(n,k) is defined for n,k >= 0.  The triangle contains only the terms with k < n.  T(n,k) = 0 for k >= n.

LINKS

Alois P. Heinz, Rows n = 1..200, flattened

FORMULA

T(n,k) = A318757(n,k) - A318757(n,k-1) for k > 0, A(n,0) = A063524(n).

EXAMPLE

Triangle T(n,k) begins:

  1;

  0,   1;

  0,   1,   1;

  0,   2,   1,   1;

  0,   3,   4,   1,  1;

  0,   6,   9,   3,  1,  1;

  0,  12,  22,   9,  3,  1, 1;

  0,  25,  54,  23,  8,  3, 1, 1;

  0,  52, 138,  60, 23,  8, 3, 1, 1;

  0, 113, 346, 164, 61, 22, 8, 3, 1, 1;

MAPLE

h:= proc(n, m, t, k) option remember; `if`(m=0, binomial(n+t, t),

      `if`(n=0, 0, add(h(n-1, m-j, t+1, k), j=1..min(k, m))))

    end:

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(b(n-i*j, i-1, k)*h(A(i, k), j, 0, k), j=0..n/i)))

    end:

A:= (n, k)-> `if`(n<2, n, b(n-1$2, k)):

T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):

seq(seq(T(n, k), k=0..n-1), n=1..14);

MATHEMATICA

h[n_, m_, t_, k_] := h[n, m, t, k] = If[m == 0, Binomial[n + t, t], If[n == 0, 0, Sum[h[n - 1, m - j, t + 1, k], {j, 1, Min[k, m]}]]];

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k]*h[A[i, k], j, 0, k], {j, 0, n/i}]]];

A[n_, k_] := If[n < 2, n, b[n - 1, n - 1, k]];

T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];

Table[T[n, k], {n, 1, 14}, {k, 0, n - 1}] // Flatten (* Jean-Fran├žois Alcover, May 11 2019, after Alois P. Heinz *)

CROSSREFS

Columns k=0-10 give: A063524, A004111 (for n>1), A318859, A318860, A318861, A318862, A318863, A318864, A318865, A318866, A318867.

Row sums give A000081.

T(2n+2,n+1) gives A255705.

Cf. A318754, A318757.

Sequence in context: A124816 A238349 A318754 * A334192 A124790 A325734

Adjacent sequences:  A318755 A318756 A318757 * A318759 A318760 A318761

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 02 2018

STATUS

approved

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Last modified August 13 12:57 EDT 2022. Contains 356091 sequences. (Running on oeis4.)