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A116853 Difference triangle of factorial numbers read by upward diagonals.

%I A116853 #25 Dec 21 2019 15:09:00
%S A116853 1,1,2,3,4,6,11,14,18,24,53,64,78,96,120,309,362,426,504,600,720,2119,
%T A116853 2428,2790,3216,3720,4320,5040,16687,18806,21234,24024,27240,30960,
%U A116853 35280,40320
%N A116853 Difference triangle of factorial numbers read by upward diagonals.
%C A116853 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 A116853 Reinhard Zumkeller, <a href="/A116853/b116853.txt">Rows n = 1..125 of triangle, flattened</a>
%F A116853 Take successive difference rows of factorial numbers n! starting with n=1. Reorient into a triangle format.
%e A116853 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 A116853 [1]    1;
%e A116853 [2]    1,   2;
%e A116853 [3]    3,   4,   6;
%e A116853 [4]   11,  14,  18,  24;
%e A116853 [5]   53,  64,  78,  96, 120;
%e A116853 [6]  309, 362, 426, 504, 600, 720;
%e A116853 ...
%t A116853 rows = 8;
%t A116853 rr = Range[rows]!;
%t A116853 dd = Table[Differences[rr, n], {n, 0, rows-1}];
%t A116853 T = Array[t, {rows, rows}];
%t A116853 Do[Thread[Evaluate[Diagonal[T, -k+1]] = dd[[k, ;;rows-k+1]]], {k, rows}];
%t A116853 Table[t[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Dec 21 2019 *)
%o A116853 (Haskell)
%o A116853 a116853 n k = a116853_tabl !! (n-1) !! (k-1)
%o A116853 a116853_row n = a116853_tabl !! (n-1)
%o A116853 a116853_tabl = map reverse $ f (tail a000142_list) [] where
%o A116853    f (u:us) vs = ws : f us ws where ws = scanl (-) u vs
%o A116853 -- _Reinhard Zumkeller_, Aug 31 2014
%Y A116853 Cf. A000142 (factorial numbers).
%Y A116853 Cf. A000255 (first column and inverse binomial transform of A000142).
%Y A116853 N-th forward differences of A000142: A001563 (1st), A001564 (2nd), A001565 (3rd), A001688 (4th), A001689 (5th).
%Y A116853 Cf. A047920 (with 0!, different order), A068106 (with 0!), A180191 (row sums), A246606 (central terms).
%K A116853 nonn,tabl
%O A116853 1,3
%A A116853 _Gary W. Adamson_, Feb 24 2006