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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

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

Project Euler, Problem 427: n-sequences

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-Fran├žois 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 A196061

Adjacent sequences:  A228151 A228152 A228153 * A228155 A228156 A228157

KEYWORD

nonn,tabl

AUTHOR

Walt Rorie-Baety, Aug 15 2013

STATUS

approved

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Last modified November 21 22:16 EST 2019. Contains 329383 sequences. (Running on oeis4.)