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

1,4

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

Links

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

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).

Example

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).

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 *)

Prog

(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

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).

Sequence in context: A126226 A307665 A144156 * A331692 A016567 A304058

Adjacent sequences: A116851 A116852 A116853 * A116855 A116856 A116857

Keyword

nonn,easy,tabl

Author

Gary W. Adamson, Feb 24 2006

Extensions

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

Status

approved