A228154 T(n,k) is the number of s in {1,...,n}^n having longest contiguous subsequence with the same value of length k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 2, 2, 12, 12, 3, 108, 120, 24, 4, 1280, 1520, 280, 40, 5, 18750, 23400, 3930, 510, 60, 6, 326592, 423360, 65016, 7644, 840, 84, 7, 6588344, 8800008, 1241464, 132552, 13440, 1288, 112, 8, 150994944, 206622720, 26911296, 2622528, 244944, 22032, 1872, 144, 9

Offset

1,2

Links

Alois P. Heinz, Rows n = 1..141, flattened

Project Euler, Problem 427: n-sequences

Formula

Sum_{k=1..n} k * T(n,k) = A228194(n). - Alois P. Heinz, Dec 23 2020

Example

T(1,1) =  1: [1].
T(2,1) =  2: [1,2], [2,1].
T(2,2) =  2: [1,1], [2,2].
T(3,1) = 12: [1,2,1], [1,2,3], [1,3,1], [1,3,2], [2,1,2], [2,1,3], [2,3,1], [2,3,2], [3,1,2], [3,1,3], [3,2,1], [3,2,3].
T(3,2) = 12: [1,1,2], [1,1,3], [1,2,2], [1,3,3], [2,1,1], [2,2,1], [2,2,3], [2,3,3], [3,1,1], [3,2,2], [3,3,1], [3,3,2].
T(3,3) =  3: [1,1,1], [2,2,2], [3,3,3].
Triangle T(n,k) begins:
.       1;
.       2,       2;
.      12,      12,       3;
.     108,     120,      24,      4;
.    1280,    1520,     280,     40,     5;
.   18750,   23400,    3930,    510,    60,    6;
.  326592,  423360,   65016,   7644,   840,   84,   7;
. 6588344, 8800008, 1241464, 132552, 13440, 1288, 112,  8;

Maple

T:= proc(n) option remember; local b; b:=
      proc(m, s, i) option remember; `if`(m>i or s>m, 0,
        `if`(i=1, n, `if`(s=1, (n-1)*add(b(m, h, i-1), h=1..m),
         b(m, s-1, i-1) +`if`(s=m, b(m-1, s-1, i-1), 0))))
      end; forget(b);
      seq(add(b(k, s, n), s=1..k), k=1..n)
    end:
seq(T(n), n=1..12);  # Alois P. Heinz, Aug 18 2013

Mathematica

T[n_] := T[n] = Module[{b}, b[m_, s_, i_] := b[m, s, i] = If[m>i || s>m, 0, If[i == 1, n, If[s == 1, (n-1)*Sum[b[m, h, i-1], {h, 1, m}], b[m, s-1, i-1] + If[s == m, b[m-1, s-1, i-1], 0]]]]; Table[Sum[b[k, s, n], {s, 1, k}], {k, 1, n}]]; Table[ T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Mar 06 2015, after Alois P. Heinz *)

Crossrefs

Row sums give: A000312.

Column k=1 gives: A055897.

Main diagonal gives: A000027.

Lower diagonal gives: 2*A180291.

Cf. A228194, A228273, A228617.

Sequence in context: A307659 A327874 A190295 * A275279 A109767 A339297

Adjacent sequences: A228151 A228152 A228153 * A228155 A228156 A228157

Keyword

nonn,tabl

Author

Walt Rorie-Baety, Aug 15 2013

Status

approved