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

0,6

Comments

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)

Links

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

Formula

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

Example

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.

Maple

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

Mathematica

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 *)

Prog

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

Crossrefs

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.

Sequence in context: A321969 A163770 A035561 * A186964 A005856 A157876

Adjacent sequences: A068103 A068104 A068105 * A068107 A068108 A068109

Keyword

nonn,easy,tabl,nice

Author

N. J. A. Sloane, Apr 12 2002

Extensions

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

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

Status

approved