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%I #36 Oct 22 2019 17:55:39
%S 0,0,0,0,1,0,0,2,2,0,0,3,6,3,0,0,4,14,12,4,0,0,5,30,39,20,5,0,0,6,62,
%T 120,84,30,6,0,0,7,126,363,340,155,42,7,0,0,8,254,1092,1364,780,258,
%U 56,8,0,0,9,510,3279,5460,3905,1554,399,72,9,0
%N A(n,k) = Sum_{i=1..k} n^i; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A(n,k) is the total sum of lengths of longest ending contiguous subsequences with the same value over all s in {1,...,n}^k:
%C A(4,1) = 4 = 1+1+1+1: [1], [2], [3], [4].
%C A(1,4) = 4: [1,1,1,1].
%C 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].
%C A(2,3) = 14 = 3+1+1+2+2+1+1+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].
%H Alois P. Heinz, <a href="/A228275/b228275.txt">Antidiagonals n = 0..140, flattened</a>
%F A(1,k) = k, else A(n,k) = n/(n-1)*(n^k-1).
%F A(n,k) = Sum_{i=1..k} n^i.
%F A(n,k) = Sum_{i=1..k+1} binomial(k+1,i)*A(n-i,k)*(-1)^(i+1) for n>k, given values A(0,k), A(1,k),..., A(k,k). - _Yosu Yurramendi_, Sep 03 2013
%e Square array A(n,k) begins:
%e 0, 0, 0, 0, 0, 0, 0, 0, ...
%e 0, 1, 2, 3, 4, 5, 6, 7, ...
%e 0, 2, 6, 14, 30, 62, 126, 254, ...
%e 0, 3, 12, 39, 120, 363, 1092, 3279, ...
%e 0, 4, 20, 84, 340, 1364, 5460, 21844, ...
%e 0, 5, 30, 155, 780, 3905, 19530, 97655, ...
%e 0, 6, 42, 258, 1554, 9330, 55986, 335922, ...
%e 0, 7, 56, 399, 2800, 19607, 137256, 960799, ...
%p A:= (n, k)-> `if`(n=1, k, (n/(n-1))*(n^k-1)):
%p seq(seq(A(n, d-n), n=0..d), d=0..12);
%t a[0, 0] = 0; a[1, k_] := k; a[n_, k_] := n*(n^k-1)/(n-1); Table[a[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Dec 16 2013 *)
%Y Columns k=0-10 give: A000004, A001477, A002378, A027444, A027445, A152031, A228290, A228291, A228292, A228293, A228294.
%Y Rows n=0-11 give: A000004, A001477, A000918(k+1), A029858(k+1), A080674, A104891, A105281, A104896, A052379(k-1), A052386, A105279, A105280.
%Y Main diagonal gives A031972.
%Y Lower diagonal gives A226238.
%Y Cf. A228250.
%K nonn,tabl,easy
%O 0,8
%A _Alois P. Heinz_, Aug 19 2013