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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.

%I A228154 #42 Dec 23 2020 10:05:53
%S A228154 1,2,2,12,12,3,108,120,24,4,1280,1520,280,40,5,18750,23400,3930,510,
%T A228154 60,6,326592,423360,65016,7644,840,84,7,6588344,8800008,1241464,
%U A228154 132552,13440,1288,112,8,150994944,206622720,26911296,2622528,244944,22032,1872,144,9
%N 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.
%H A228154 Alois P. Heinz, <a href="/A228154/b228154.txt">Rows n = 1..141, flattened</a>
%H A228154 Project Euler, <a href="http://projecteuler.net/problem=427">Problem 427: n-sequences</a>
%F A228154 Sum_{k=1..n} k * T(n,k) = A228194(n). - _Alois P. Heinz_, Dec 23 2020
%e A228154 T(1,1) =  1: [1].
%e A228154 T(2,1) =  2: [1,2], [2,1].
%e A228154 T(2,2) =  2: [1,1], [2,2].
%e A228154 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].
%e A228154 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].
%e A228154 T(3,3) =  3: [1,1,1], [2,2,2], [3,3,3].
%e A228154 Triangle T(n,k) begins:
%e A228154 .       1;
%e A228154 .       2,       2;
%e A228154 .      12,      12,       3;
%e A228154 .     108,     120,      24,      4;
%e A228154 .    1280,    1520,     280,     40,     5;
%e A228154 .   18750,   23400,    3930,    510,    60,    6;
%e A228154 .  326592,  423360,   65016,   7644,   840,   84,   7;
%e A228154 . 6588344, 8800008, 1241464, 132552, 13440, 1288, 112,  8;
%p A228154 T:= proc(n) option remember; local b; b:=
%p A228154       proc(m, s, i) option remember; `if`(m>i or s>m, 0,
%p A228154         `if`(i=1, n, `if`(s=1, (n-1)*add(b(m, h, i-1), h=1..m),
%p A228154          b(m, s-1, i-1) +`if`(s=m, b(m-1, s-1, i-1), 0))))
%p A228154       end; forget(b);
%p A228154       seq(add(b(k, s, n), s=1..k), k=1..n)
%p A228154     end:
%p A228154 seq(T(n), n=1..12);  # _Alois P. Heinz_, Aug 18 2013
%t A228154 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_ *)
%Y A228154 Row sums give: A000312.
%Y A228154 Column k=1 gives: A055897.
%Y A228154 Main diagonal gives: A000027.
%Y A228154 Lower diagonal gives: 2*A180291.
%Y A228154 Cf. A228194, A228273, A228617.
%K A228154 nonn,tabl
%O A228154 1,2
%A A228154 _Walt Rorie-Baety_, Aug 15 2013