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First differences of the rows in the triangle of A116853, starting with 0.
2

%I #19 Jul 01 2022 07:25:24

%S 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 426,504,600,720,16687,2119,2428,2790,3216,3720,4320,5040,148329,

%U 16687,18806,21234,24024,27240,30960,35280,40320,1468457,148329,165016,183822,205056,229080,256320,287280,322560,362880

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

%C Row n contains the first differences of row n of A116853, starting with T(n,1) = A116853(n,1) - 0.

%C As in A116853, 0! = 1 is omitted here. - _Georg Fischer_, Mar 23 2019

%H Reinhard Zumkeller, <a href="/A116854/b116854.txt">Rows n = 1..125 of triangle, flattened</a>

%F T(n,k) = A116853(n,k) - A116853(n,k-1) if k>1.

%F T(n,1) = A116853(n,1) = A000255(n-1).

%F Sum_{k=1..n} T(n,1) = n! = A000142(n).

%e First few rows of the triangle are:

%e [1] 1;

%e [2] 1, 1;

%e [3] 3, 1, 2;

%e [4] 11, 3, 4, 6;

%e [5] 53, 11, 14, 18, 24;

%e [6] 309, 53, 64, 78, 96, 120;

%e [7] 2119, 309, 362, 426, 504, 600, 720;

%e ...

%e For example, row 4 (11, 3, 4, 6) are first differences along row 4 of A116853: ((0), 11, 14, 18, 24).

%p 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 := proc(n,k) if k = 1 then A116853(n,1) ; else A116853(n,k) -A116853(n,k-1) ; end if; end proc:

%p seq(seq(A116854(n,k),k=1..n),n=1..15) ; # _R. J. Mathar_, Mar 27 2010

%t rows = 10;

%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[({0}~Join~Table[t[n, k], {k, 1, n}]) // Differences, {n, 1, rows}] // Flatten (* _Jean-François Alcover_, Dec 21 2019 *)

%o (Haskell)

%o a116854 n k = a116854_tabl !! (n-1) !! (k-1)

%o a116854_row n = a116854_tabl !! (n-1)

%o a116854_tabl = [1] : zipWith (:) (tail $ map head tss) tss

%o where tss = a116853_tabl

%o -- _Reinhard Zumkeller_, Aug 31 2014

%Y Cf. A000255, A116853.

%Y 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 nonn,easy,tabl

%O 1,4

%A _Gary W. Adamson_, Feb 24 2006

%E Definition made concrete and sequence extended by _R. J. Mathar_, Mar 27 2010