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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-Fran├žois 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

AUTHOR

Alois P. Heinz, Aug 19 2013

STATUS

approved

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Last modified February 20 15:02 EST 2019. Contains 320327 sequences. (Running on oeis4.)