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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). 21
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 (list; table; graph; refs; listen; history; text; internal format)
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.

E. 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, 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

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 include A000166, A000255, A055790, A000142.

See A047920 and A086764 for other versions.

When seen as an array, main diagonal is A033815.

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

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Last modified November 18 10:08 EST 2019. Contains 329261 sequences. (Running on oeis4.)