A104891 -id:A104891 - cp's OEIS frontend

A228275 A(n,k) = Sum_{i=1..k} n^i; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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

Offset: 0

Alois P. Heinz, Aug 19 2013

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].

			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, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

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.

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);

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 *)

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

A178671 a(n) = 5^n - 5.

Original entry on oeis.org

-4, 0, 20, 120, 620, 3120, 15620, 78120, 390620, 1953120, 9765620, 48828120, 244140620, 1220703120, 6103515620, 30517578120, 152587890620, 762939453120, 3814697265620, 19073486328120, 95367431640620, 476837158203120, 2384185791015620, 11920928955078120

Offset: 0

Vincenzo Librandi, Dec 25 2010

			a(n) = A178676(n)-10 = A242329(n)-9 = A242328(n)-7 = A034474(n)-6 = A000351(n)-5. - _Elmo R. Oliveira_, Dec 06 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000351, A034474, A178676, A242328, A242329.

List([0..30], n -> 5^n-5); # G. C. Greubel, Jan 28 2019

Magma
```
[5^n-5: n in [0..25]];
```

5^Range[0,30]-5 (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)
LinearRecurrence[{6,-5},{-4,0},30] (* Harvey P. Dale, Aug 23 2024 *)

vector(30, n, n--; 5^n-5) \\ G. C. Greubel, Jan 28 2019

[5^n-5 for n in range(30)] # G. C. Greubel, Jan 28 2019

a(n) = 5*a(n-1) + 20 with a(0) = -4.
From R. J. Mathar, Jan 03 2011: (Start)
G.f.: 4*(-1+6*x)/((1-5*x)*(1-x)).
a(n) = 4*A104891(n-1), n > 0. (End)
a(n) = 6*a(n-1) - 5*a(n-2) for n > 1. - Vincenzo Librandi, Jan 25 2013
E.g.f.: exp(5*x) - 5*exp(x). - G. C. Greubel, Jan 28 2019

cp's OEIS Frontend

A228275 A(n,k) = Sum_{i=1..k} n^i; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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