OFFSET
1,2
COMMENTS
For the five sailors and one monkey problem see A254029.
The rows s of the array A give the positive solutions to the following problem: Recurrence co(k) = ((s-1)/s)*(co(k-1) - 1), for k >= 0, with co(0) = a, and the requirement c0(s) - 1 == 0 (mod s), for s >= 1. Then a = a(s, n) = A(s, n), for n >= 1.
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened).
FORMULA
T(n, k) = A(k, n - k + 1), with the array A(s, n) = n*s^(s+1) - (s - 1), for s >= 1 and n >= 1. (Array read by antidiagonals downwards.)
T(n, k) = (n - k + 1)*k^(k+1) - (k - 1), for k = 1, 2, ..., n.
O.g.f. for row s of array A: (x/(1 - x)^2)*(s^(s + 1) - (s - 1)*(1 - x)).
E.g.f. for column n of array A: n*(-W(-x)/(1 - (-W(-x)))^3) - (1 - (1 - x)*exp(x)), with the principal branch of Lambert's W-function
EXAMPLE
The array A begins:
s\n 1 2 3 4 5 6 7 8 9 ...
---------------------------------------------------------------------------
1: 1 2 3 4 5 6 7 8 9 ...
2: 7 15 23 31 39 47 55 63 71 ...
3: 79 160 241 322 403 484 565 646 727 ...
4: 1021 2045 3069 4093 5117 6141 7165 8189 9213 ...
5: 15621 31246 46871 62496 78121 93746 109371 124996 140621 ...
6: 279931 559867 839803 1119739 1399675 1679611 1959547 2239483 2519419 ...
...
s = 7: 5764795 11529596 17294397 23059198 28823999 34588800 40353601 46118402 51883203 57648004, ...
...
-----------------------------------------------------------------------------
The triangle begins:
n\k 1 2 3 4 5 6 7 8 9 10
---------------------------------------------------------------------------
1: 1
2: 2 7
3: 3 15 79
4 4 23 160 1021
5: 5 31 241 2045 15621
6: 6 39 322 3069 31246 279931
7: 7 47 403 4093 46871 559867 5764795
8: 8 55 484 5117 62496 839803 11529596 134217721
9: 9 63 565 6141 78121 1119739 17294397 268435449 3486784393
10: 10 71 646 7165 93746 1399675 23059198 402653177 6973568794 99999999991
...
-----------------------------------------------------------------------------
MATHEMATICA
A362359row[n_]:=Array[(n-#+1)#^(#+1)-#+1&, n]; Array[A362359row, 10] (* Paolo Xausa, Nov 17 2023 *)
CROSSREFS
KEYWORD
AUTHOR
Richard S. Fischer and Wolfdieter Lang, Jun 20 2023
STATUS
approved