A277563 -id:A277563 - cp's OEIS frontend

A068106 Euler's difference table: triangle read by rows, formed by starting with factorial numbers (A000142) and repeatedly taking differences. T(n,n) = n!, T(n,k) = T(n,k+1) - T(n-1,k).

1, 0, 1, 1, 1, 2, 2, 3, 4, 6, 9, 11, 14, 18, 24, 44, 53, 64, 78, 96, 120, 265, 309, 362, 426, 504, 600, 720, 1854, 2119, 2428, 2790, 3216, 3720, 4320, 5040, 14833, 16687, 18806, 21234, 24024, 27240, 30960, 35280, 40320, 133496, 148329, 165016, 183822, 205056, 229080, 256320, 287280, 322560, 362880

Offset: 0

N. J. A. Sloane, Apr 12 2002

Triangle T(n,k) (n >= 1, 1 <= k <= n) giving number of ways of winning with (n-k+1)st card in the generalized "Game of Thirteen" with n cards.
From Emeric Deutsch, Apr 21 2009: (Start)
T(n-1,k-1) is the number of non-derangements of {1,2,...,n} having largest fixed point equal to k. Example: T(3,1)=3 because we have 1243, 4213, and 3241.
Mirror image of A047920.
(End)

			Triangle begins:
[0]    1;
[1]    0,    1;
[2]    1,    1,    2;
[3]    2,    3,    4,    6;
[4]    9,   11,   14,   18,   24;
[5]   44,   53,   64,   78,   96,  120;
[6]  265,  309,  362,  426,  504,  600,  720;
[7] 1854, 2119, 2428, 2790, 3216, 3720, 4320, 5040.

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
W. Y. C. Chen et al., Higher-order log-concavity in Euler's difference table, Discrete Math., 311 (2011), 2128-2134.
P. R. de Montmort, On the Game of Thirteen (1713), reprinted in Annotated Readings in the History of Statistics, ed. H. A. David and A. W. F. Edwards, Springer-Verlag, 2001, pp. 25-29.
Emeric Deutsch and S. Elizalde, The largest and the smallest fixed points of permutations, arXiv:0904.2792 [math.CO], 2009.
D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.
Philip Feinsilver and John McSorley, Zeons, Permanents, the Johnson scheme, and Generalized Derangements, arXiv:1710.00788 [math.CO], (2017); see page 29.
P. Feinsilver and J. McSorley, Zeons, Permanents, the Johnson scheme, and Generalized Derangements, International Journal of Combinatorics, 2011 (2011).
Fanja Rakotondrajao, k-Fixed-Points-Permutations, Integers: Electronic journal of combinatorial number theory 7 (2007) A36.
Index entries for sequences related to factorial numbers

Row sums give A002467.
Diagonals give A000142, A001563, A001564, A001565, A001688, A001689, A023043, A023044, A023045, A023046, A023047 (factorials and k-th differences, k=1..10).
See A047920 and A086764 for other versions.
T(2*n, n) is A033815.
Columns k=0..10 give A000166, A000255, A055790, A277609, A277563, A280425, A280920, A284204, A284205, A284206, A284207.

a068106 n k = a068106_tabl !! n !! k
a068106_row n = a068106_tabl !! n
a068106_tabl = map reverse a047920_tabl
-- Reinhard Zumkeller, Mar 05 2012

d[0] := 1: for n to 15 do d[n] := n*d[n-1]+(-1)^n end do: T := proc (n, k) if k <= n then sum(binomial(k, j)*d[n-j], j = 0 .. k) else 0 end if end proc: for n from 0 to 9 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form; Emeric Deutsch, Jul 18 2009

t[n_, k_] := Sum[(-1)^j*Binomial[n-k, j]*(n-j)!, {j, 0, n}]; Flatten[ Table[ t[n, k], {n, 0, 9}, {k, 0, n}]] (* Jean-François Alcover, Feb 21 2012, after Philippe Deléham *)
T[n_, k_] := n! HypergeometricPFQ[{k-n}, {-n}, -1];
Table[T[n, k], {n,0,9}, {k,0,n}] // Flatten (* Peter Luschny, Oct 05 2017 *)

T(n, k) = Sum_{j>= 0} (-1)^j*binomial(n-k, j)*(n-j)!. - Philippe Deléham, May 29 2005
From Emeric Deutsch, Jul 18 2009: (Start)
T(n,k) = Sum_{j=0..k} d(n-j)*binomial(k, j), where d(i) = A000166(i) are the derangement numbers.
Sum_{k=0..n} (k+1)*T(n,k) = A000166(n+2) (the derangement numbers). (End)
T(n, k) = n!*hypergeom([k-n], [-n], -1). - Peter Luschny, Oct 05 2017
D-finite recurrence for columns: T(n,k) = n*T(n-1,k) + (n-k)*T(n-2,k). - Georg Fischer, Aug 13 2022

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 01 2003

Edited by N. J. A. Sloane, Sep 24 2011

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

Original entry on oeis.org

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

Gary W. Adamson, Feb 24 2006

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

			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).

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

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).

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

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

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 *)

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).

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

cp's OEIS Frontend

A068106 Euler's difference table: triangle read by rows, formed by starting with factorial numbers (A000142) and repeatedly taking differences. T(n,n) = n!, T(n,k) = T(n,k+1) - T(n-1,k).

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