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A108283 Triangle read by rows, generated from (..., 3, 2, 1).

%I A108283 #20 Feb 07 2022 08:17:06
%S A108283 1,1,3,1,5,6,1,7,17,10,1,9,34,49,15,1,11,57,142,129,21,1,13,86,313,
%T A108283 547,321,28,1,15,121,586,1593,2005,769,36,1,17,162,985,3711,7737,7108,
%U A108283 1793,45,1,19,209,1534,7465,22461,36409,24604,4097,55,1,21,262,2257,13539,54121,131836,167481,83653,9217,66
%N A108283 Triangle read by rows, generated from (..., 3, 2, 1).
%C A108283 Inverse binomial transforms of each column form the rows of A108284. Rightmost diagonal = triangular numbers, (A000217); while diagonals going to the left from (1, 3, 6, ...) are A000337 starting with 1: (1, 5, 17, 49, ...); A014915: (1, 7, 34, 142, ...); A014916: (1, 9, 57, ...); A014917: (1, 11, 86, ...).
%F A108283 n-th column = f(x), x = 1, 2, 3; n*x^(n-1) + (n-1)*x^(n-2) + (n-3)*x^(n-3) + ... + 1.
%F A108283 T(n,k) = (1+ (n-k+1)^k*(n*k-k^2-1))/ (n-k)^2, n>k. - _Jean-François Alcover_, Sep 13 2016
%e A108283 4th column = 10, 49, 142, 313, ... = f(x), x = 1, 2, 3; 4x^3 + 3x^2 + 2x + 1. f(3) = 142.
%e A108283 First few rows of the triangle:
%e A108283   1;
%e A108283   1,  3;
%e A108283   1,  5,  6;
%e A108283   1,  7, 17,  10;
%e A108283   1,  9, 34,  49,  15;
%e A108283   1, 11, 57, 142, 129, 21;
%e A108283   ...
%p A108283 A108283 := proc(n,k)
%p A108283     local x ;
%p A108283     x := n-k+1 ;
%p A108283     add( i*x^(i-1),i=1..k) ;
%p A108283 end proc:
%p A108283 seq(seq( A108283(n,k),k=1..n),n=1..10) ; # _R. J. Mathar_, Sep 14 2016
%t A108283 T[_, 1] := 1; T[n_, n_] := n (n + 1)/2; T[n_, k_] := (1 - (n - k + 1)^k*(k^2 - k*n + 1))/(n - k)^2; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 13 2016 *)
%Y A108283 Cf. A059045, A108284, A000217, A000337, A014915, A014916, A014917.
%K A108283 nonn,tabl,easy
%O A108283 1,3
%A A108283 _Gary W. Adamson_, May 30 2005
%E A108283 More terms from _Jean-François Alcover_, Sep 13 2016