The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A318758 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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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.
(history; published version)

#18 by Susanna Cuyler at Sat May 11 10:11:19 EDT 2019

STATUS

reviewed

approved

#17 by Wesley Ivan Hurt at Sat May 11 06:10:44 EDT 2019

STATUS

proposed

reviewed

#16 by Jean-François Alcover at Sat May 11 05:37:33 EDT 2019

STATUS

editing

proposed

#15 by Jean-François Alcover at Sat May 11 05:37:28 EDT 2019

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-François Alcover, May 11 2019, after Alois P. Heinz *)

STATUS

approved

editing

#14 by Alois P. Heinz at Tue Sep 04 18:52:38 EDT 2018

STATUS

editing

approved

#13 by Alois P. Heinz at Tue Sep 04 18:25:46 EDT 2018

CROSSREFS

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

#12 by Alois P. Heinz at Tue Sep 04 18:17:00 EDT 2018

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);

#11 by Alois P. Heinz at Tue Sep 04 16:09:33 EDT 2018

CROSSREFS

Columns k=0-1 give: A063524, A004111 (for n>1).

#10 by Alois P. Heinz at Tue Sep 04 16:05:42 EDT 2018

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;

#9 by Alois P. Heinz at Tue Sep 04 16:04:53 EDT 2018

NAME

T2

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.