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A228250 Total sum A(n,k) of lengths of longest contiguous subsequences with the same value over all s in {1,...,n}^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A228250 #24 Sep 21 2018 22:30:05
%S A228250 0,0,0,0,1,0,0,2,2,0,0,3,6,3,0,0,4,16,12,4,0,0,5,38,45,20,5,0,0,6,86,
%T A228250 156,96,30,6,0,0,7,188,519,436,175,42,7,0,0,8,404,1680,1916,980,288,
%U A228250 56,8,0,0,9,856,5349,8232,5345,1914,441,72,9,0
%N A228250 Total sum A(n,k) of lengths of longest contiguous subsequences with the same value over all s in {1,...,n}^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A228250 Alois P. Heinz, <a href="/A228250/b228250.txt">Antidiagonals n = 0..140, flattened</a>
%H A228250 Project Euler, <a href="https://projecteuler.net/problem=427">Problem 427: n-sequences</a>
%e A228250 A(4,1) = 4 = 1+1+1+1: [1], [2], [3], [4].
%e A228250 A(1,4) = 4: [1,1,1,1].
%e A228250 A(3,2) = 12 = 2+1+1+1+2+1+1+1+2: [1,1], [1,2], [1,3], [2,1], [2,2], [2,3], [3,1], [3,2], [3,3].
%e A228250 A(2,3) = 16 = 3+2+1+2+2+1+2+3: [1,1,1], [1,1,2], [1,2,1], [1,2,2], [2,1,1], [2,1,2], [2,2,1], [2,2,2].
%e A228250 Square array A(n,k) begins:
%e A228250   0, 0,  0,   0,    0,     0,      0,       0, ...
%e A228250   0, 1,  2,   3,    4,     5,      6,       7, ...
%e A228250   0, 2,  6,  16,   38,    86,    188,     404, ...
%e A228250   0, 3, 12,  45,  156,   519,   1680,    5349, ...
%e A228250   0, 4, 20,  96,  436,  1916,   8232,   34840, ...
%e A228250   0, 5, 30, 175,  980,  5345,  28610,  151115, ...
%e A228250   0, 6, 42, 288, 1914, 12450,  79716,  504492, ...
%e A228250   0, 7, 56, 441, 3388, 25571, 190428, 1403689, ...
%p A228250 b:= proc(n, m, s, i) option remember; `if`(m>i or s>m, 0,
%p A228250       `if`(i=0, 1, `if`(i=1, n, `if`(s=1, (n-1)*add(
%p A228250          b(n, m, h, i-1), h=1..m), b(n, m, s-1, i-1)+
%p A228250       `if`(s=m, b(n, m-1, s-1, i-1), 0)))))
%p A228250     end:
%p A228250 A:= (n, k)-> add(m*add(b(n, m, s, k), s=1..m), m=1..k):
%p A228250 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A228250 b[n_, m_, s_, i_] := b[n, m, s, i] = If[m>i || s>m, 0, If[i == 0, 1, If[i == 1, n, If[s == 1, (n-1)*Sum[b[n, m, h, i-1], {h, 1, m}], b[n, m, s-1, i-1] + If[s == m, b[n, m-1, s-1, i-1], 0]]]]]; A[n_, k_] := Sum[m*Sum[b[n, m, s, k], {s, 1, m}], {m, 1, k}]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Jan 19 2015, after _Alois P. Heinz_ *)
%Y A228250 Columns k=0-3 give: A000004, A001477, A002378, A152618(n+1).
%Y A228250 Rows n=0-2 give: A000004, A001477, 2*A102712.
%Y A228250 Main diagonal gives: A228194.
%Y A228250 Cf. A228275.
%K A228250 nonn,tabl
%O A228250 0,8
%A A228250 _Alois P. Heinz_, Aug 18 2013