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%I #25 Dec 21 2019 15:09:00
%S 1,1,2,3,4,6,11,14,18,24,53,64,78,96,120,309,362,426,504,600,720,2119,
%T 2428,2790,3216,3720,4320,5040,16687,18806,21234,24024,27240,30960,
%U 35280,40320
%N Difference triangle of factorial numbers read by upward diagonals.
%C This is a subsequence of Euler's difference table A068106 and of A047920 (in a different ordering), since 0! = 1 was left out here. - _Georg Fischer_, Mar 23 2019
%H Reinhard Zumkeller, <a href="/A116853/b116853.txt">Rows n = 1..125 of triangle, flattened</a>
%F Take successive difference rows of factorial numbers n! starting with n=1. Reorient into a triangle format.
%e Starting with 1, 2, 6, 24, 120 ... we take the first difference row (A001563), second, third, etc. Reorient into a flush left format, getting:
%e [1] 1;
%e [2] 1, 2;
%e [3] 3, 4, 6;
%e [4] 11, 14, 18, 24;
%e [5] 53, 64, 78, 96, 120;
%e [6] 309, 362, 426, 504, 600, 720;
%e ...
%t rows = 8;
%t rr = Range[rows]!;
%t dd = Table[Differences[rr, n], {n, 0, rows-1}];
%t T = Array[t, {rows, rows}];
%t Do[Thread[Evaluate[Diagonal[T, -k+1]] = dd[[k, ;;rows-k+1]]], {k, rows}];
%t Table[t[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Dec 21 2019 *)
%o (Haskell)
%o a116853 n k = a116853_tabl !! (n-1) !! (k-1)
%o a116853_row n = a116853_tabl !! (n-1)
%o a116853_tabl = map reverse $ f (tail a000142_list) [] where
%o f (u:us) vs = ws : f us ws where ws = scanl (-) u vs
%o -- _Reinhard Zumkeller_, Aug 31 2014
%Y Cf. A000142 (factorial numbers).
%Y Cf. A000255 (first column and inverse binomial transform of A000142).
%Y N-th forward differences of A000142: A001563 (1st), A001564 (2nd), A001565 (3rd), A001688 (4th), A001689 (5th).
%Y Cf. A047920 (with 0!, different order), A068106 (with 0!), A180191 (row sums), A246606 (central terms).
%K nonn,tabl
%O 1,3
%A _Gary W. Adamson_, Feb 24 2006
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