The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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. 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified February 17 18:14 EST 2020. Contains 332005 sequences. (Running on oeis4.)