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A068106
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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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9
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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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.
Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), 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)
Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 18 2009: (Start)
Sum[(k+1)*T(n,k), k=0..n]=A000166(n+2) (the derangement numbers).
(End)
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REFERENCES
| 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:0904.2792v1, 2009. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 21 2009]
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LINKS
| D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.
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FORMULA
| T(n, k) = Sum_{ j>= 0} (-1)^j*binomial(n-k, j)*(n-j)! . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 29 2005
Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 18 2009: (Start)
T(n,k)=Sum[d(n-j)*binom(k, j), j=0..k], where d(i)=A000166(i) are the derangement numbers.
(End)
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EXAMPLE
| Triangle begins
1..1..2..6..24..120..720...
...0..1..4..18..96...600...
......1..3..14..78...504...
.........2..11..64...426...
............9...53...362...
................44...309...
.....................265...
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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 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 18 2009]
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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.
Cf. A047920 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 21 2009]
Sequence in context: A068598 A163770 A035561 * A186964 A005856 A157876
Adjacent sequences: A068103 A068104 A068105 * A068107 A068108 A068109
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KEYWORD
| nonn,easy,tabl,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Apr 12 2002
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EXTENSIONS
| More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 01 2003
Edited by N. J. A. Sloane, Sep 24 2011
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