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A116854 First differences of the rows in the triangle of A116853, starting with 0.

%I A116854 #19 Jul 01 2022 07:25:24
%S A116854 1,1,1,3,1,2,11,3,4,6,53,11,14,18,24,309,53,64,78,96,120,2119,309,362,
%T A116854 426,504,600,720,16687,2119,2428,2790,3216,3720,4320,5040,148329,
%U A116854 16687,18806,21234,24024,27240,30960,35280,40320,1468457,148329,165016,183822,205056,229080,256320,287280,322560,362880
%N A116854 First differences of the rows in the triangle of A116853, starting with 0.
%C A116854 Row n contains the first differences of row n of A116853, starting with T(n,1) = A116853(n,1) - 0.
%C A116854 As in A116853, 0! = 1 is omitted here. - _Georg Fischer_, Mar 23 2019
%H A116854 Reinhard Zumkeller, <a href="/A116854/b116854.txt">Rows n = 1..125 of triangle, flattened</a>
%F A116854 T(n,k) = A116853(n,k) - A116853(n,k-1) if k>1.
%F A116854 T(n,1) = A116853(n,1) = A000255(n-1).
%F A116854 Sum_{k=1..n} T(n,1) = n! = A000142(n).
%e A116854 First few rows of the triangle are:
%e A116854 [1]    1;
%e A116854 [2]    1,   1;
%e A116854 [3]    3,   1,   2;
%e A116854 [4]   11,   3,   4,   6;
%e A116854 [5]   53,  11,  14,  18,  24;
%e A116854 [6]  309,  53,  64,  78,  96, 120;
%e A116854 [7] 2119, 309, 362, 426, 504, 600, 720;
%e A116854 ...
%e A116854 For example, row 4 (11, 3, 4, 6) are first differences along row 4 of A116853: ((0), 11, 14, 18, 24).
%p A116854 A116853 := proc(n,k) option remember ; if n = k then n! ; else procname(n,k+1)-procname(n-1,k) ; end if; end proc:
%p A116854 A116854 := proc(n,k) if k = 1 then A116853(n,1) ; else A116853(n,k) -A116853(n,k-1) ; end if; end proc:
%p A116854 seq(seq(A116854(n,k),k=1..n),n=1..15) ; # _R. J. Mathar_, Mar 27 2010
%t A116854 rows = 10;
%t A116854 rr = Range[rows]!;
%t A116854 dd = Table[Differences[rr, n], {n, 0, rows - 1}];
%t A116854 T = Array[t, {rows, rows}];
%t A116854 Do[Thread[Evaluate[Diagonal[T, -k+1]] = dd[[k, ;; rows-k+1]]], {k, rows}];
%t A116854 Table[({0}~Join~Table[t[n, k], {k, 1, n}]) // Differences, {n, 1, rows}] // Flatten (* _Jean-François Alcover_, Dec 21 2019 *)
%o A116854 (Haskell)
%o A116854 a116854 n k = a116854_tabl !! (n-1) !! (k-1)
%o A116854 a116854_row n = a116854_tabl !! (n-1)
%o A116854 a116854_tabl = [1] : zipWith (:) (tail $ map head tss) tss
%o A116854                where tss = a116853_tabl
%o A116854 -- _Reinhard Zumkeller_, Aug 31 2014
%Y A116854 Cf. A000255, A116853.
%Y A116854 Cf. A000142 (row sums), A033815 (central terms), A047920, A068106 (with 0!), A055790 (column k=3), A277609 (k=4), A277563 (k=5), A280425 (k=6).
%K A116854 nonn,easy,tabl
%O A116854 1,4
%A A116854 _Gary W. Adamson_, Feb 24 2006
%E A116854 Definition made concrete and sequence extended by _R. J. Mathar_, Mar 27 2010