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%I #14 Nov 17 2023 08:13:58
%S 1,2,7,3,15,79,4,23,160,1021,5,31,241,2045,15621,6,39,322,3069,31246,
%T 279931,7,47,403,4093,46871,559867,5764795,8,55,484,5117,62496,839803,
%U 11529596,134217721,9,63,565,6141,78121,1119739,17294397,268435449,3486784393,10,71,646,7165,93746,1399675,23059198,402653177,6973568794,99999999991
%N Triangle T read by rows, obtained from the array A for the solutions of the Monkey and Coconuts Problem (s sailors and one coconut to the monkey).
%C For the five sailors and one monkey problem see A254029.
%C 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.
%H Paolo Xausa, <a href="/A362359/b362359.txt">Table of n, a(n) for n = 1..11325</a> (rows 1..150 of the triangle, flattened).
%F 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.)
%F T(n, k) = (n - k + 1)*k^(k+1) - (k - 1), for k = 1, 2, ..., n.
%F O.g.f. for row s of array A: (x/(1 - x)^2)*(s^(s + 1) - (s - 1)*(1 - x)).
%F 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
%e The array A begins:
%e s\n 1 2 3 4 5 6 7 8 9 ...
%e ---------------------------------------------------------------------------
%e 1: 1 2 3 4 5 6 7 8 9 ...
%e 2: 7 15 23 31 39 47 55 63 71 ...
%e 3: 79 160 241 322 403 484 565 646 727 ...
%e 4: 1021 2045 3069 4093 5117 6141 7165 8189 9213 ...
%e 5: 15621 31246 46871 62496 78121 93746 109371 124996 140621 ...
%e 6: 279931 559867 839803 1119739 1399675 1679611 1959547 2239483 2519419 ...
%e ...
%e s = 7: 5764795 11529596 17294397 23059198 28823999 34588800 40353601 46118402 51883203 57648004, ...
%e ...
%e -----------------------------------------------------------------------------
%e The triangle begins:
%e n\k 1 2 3 4 5 6 7 8 9 10
%e ---------------------------------------------------------------------------
%e 1: 1
%e 2: 2 7
%e 3: 3 15 79
%e 4 4 23 160 1021
%e 5: 5 31 241 2045 15621
%e 6: 6 39 322 3069 31246 279931
%e 7: 7 47 403 4093 46871 559867 5764795
%e 8: 8 55 484 5117 62496 839803 11529596 134217721
%e 9: 9 63 565 6141 78121 1119739 17294397 268435449 3486784393
%e 10: 10 71 646 7165 93746 1399675 23059198 402653177 6973568794 99999999991
%e ...
%e -----------------------------------------------------------------------------
%t A362359row[n_]:=Array[(n-#+1)#^(#+1)-#+1&,n];Array[A362359row,10] (* _Paolo Xausa_, Nov 17 2023 *)
%Y Rows of array A (columns of triangle T starting with index n): A000027, A004771(n-1), A362360, A362361, A254029.
%Y First column of array A (diagonal of triangle T): A014293.
%K nonn,tabl,easy
%O 1,2
%A Richard S. Fischer and _Wolfdieter Lang_, Jun 20 2023