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 A228275 A(n,k) = Sum_{i=1..k} n^i; square array A(n,k), n>=0, k>=0, read by antidiagonals. 19
 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, 120, 84, 30, 6, 0, 0, 7, 126, 363, 340, 155, 42, 7, 0, 0, 8, 254, 1092, 1364, 780, 258, 56, 8, 0, 0, 9, 510, 3279, 5460, 3905, 1554, 399, 72, 9, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS 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: A(4,1) = 4 = 1+1+1+1: [1], [2], [3], [4]. A(1,4) = 4: [1,1,1,1]. 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]. 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]. LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened FORMULA A(1,k) = k, else A(n,k) = n/(n-1)*(n^k-1). A(n,k) = Sum_{i=1..k} n^i. 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 EXAMPLE Square array A(n,k) begins:   0, 0,  0,   0,    0,     0,      0,      0, ...   0, 1,  2,   3,    4,     5,      6,      7, ...   0, 2,  6,  14,   30,    62,    126,    254, ...   0, 3, 12,  39,  120,   363,   1092,   3279, ...   0, 4, 20,  84,  340,  1364,   5460,  21844, ...   0, 5, 30, 155,  780,  3905,  19530,  97655, ...   0, 6, 42, 258, 1554,  9330,  55986, 335922, ...   0, 7, 56, 399, 2800, 19607, 137256, 960799, ... MAPLE A:= (n, k)-> `if`(n=1, k, (n/(n-1))*(n^k-1)): seq(seq(A(n, d-n), n=0..d), d=0..12); MATHEMATICA 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 *) CROSSREFS Columns k=0-10 give: A000004, A001477, A002378, A027444, A027445, A152031, A228290, A228291, A228292, A228293, A228294. Rows n=0-11 give: A000004, A001477, A000918(k+1), A029858(k+1), A080674, A104891, A105281, A104896, A052379(k-1), A052386, A105279, A105280. Main diagonal gives A031972. Lower diagonal gives A226238. Cf. A228250. Sequence in context: A271917 A185651 A265080 * A228250 A101164 A229079 Adjacent sequences:  A228272 A228273 A228274 * A228276 A228277 A228278 KEYWORD nonn,tabl,easy,changed AUTHOR Alois P. Heinz, Aug 19 2013 STATUS approved

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Last modified October 23 14:45 EDT 2019. Contains 328345 sequences. (Running on oeis4.)