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

1, 1, 1, 3, 1, 2, 11, 3, 4, 6, 53, 11, 14, 18, 24, 309, 53, 64, 78, 96, 120, 2119, 309, 362, 426, 504, 600, 720, 16687, 2119, 2428, 2790, 3216, 3720, 4320, 5040, 148329, 16687, 18806, 21234, 24024, 27240, 30960, 35280, 40320, 1468457, 148329, 165016, 183822, 205056, 229080, 256320, 287280, 322560, 362880

Offset: 1

Author

Gary W. Adamson, Feb 24 2006

Keywords

• nonn
• easy
• tabl

Comments

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

As in A116853, 0! = 1 is omitted here. - Georg Fischer, Mar 23 2019

Examples

			First few rows of the triangle are:
[1]    1;
[2]    1,   1;
[3]    3,   1,   2;
[4]   11,   3,   4,   6;
[5]   53,  11,  14,  18,  24;
[6]  309,  53,  64,  78,  96, 120;
[7] 2119, 309, 362, 426, 504, 600, 720;
...
For example, row 4 (11, 3, 4, 6) are first differences along row 4 of A116853: ((0), 11, 14, 18, 24).
		

Links

• Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Crossrefs

Cf. A000255, A116853.

Cf. A000142 (row sums), A033815 (central terms), A047920, A068106 (with 0!), A055790 (column k=3), A277609 (k=4), A277563 (k=5), A280425 (k=6).

Programs

Haskell

a116854 n k = a116854_tabl !! (n-1) !! (k-1)
a116854_row n = a116854_tabl !! (n-1)
a116854_tabl = [1] : zipWith (:) (tail $ map head tss) tss
               where tss = a116853_tabl
-- Reinhard Zumkeller, Aug 31 2014

Maple

A116853 := proc(n,k) option remember ; if n = k then n! ; else procname(n,k+1)-procname(n-1,k) ; end if; end proc:
A116854 := proc(n,k) if k = 1 then A116853(n,1) ; else A116853(n,k) -A116853(n,k-1) ; end if; end proc:
seq(seq(A116854(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Mar 27 2010

Mathematica

rows = 10;
rr = Range[rows]!;
dd = Table[Differences[rr, n], {n, 0, rows - 1}];
T = Array[t, {rows, rows}];
Do[Thread[Evaluate[Diagonal[T, -k+1]] = dd[[k, ;; rows-k+1]]], {k, rows}];
Table[({0}~Join~Table[t[n, k], {k, 1, n}]) // Differences, {n, 1, rows}] // Flatten (* Jean-François Alcover, Dec 21 2019 *)

Formula

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

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

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

Extensions

Definition made concrete and sequence extended by R. J. Mathar, Mar 27 2010