%I A068106
%S A068106 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,
%T A068106 504,600,720,1854,2119,2428,2790,3216,3720,4320,5040,14833,16687,18806,
%U A068106 21234,24024,27240,30960,35280,40320,133496,148329,165016,183822,205056
%N A068106 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).
%C A068106 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.
%C A068106 Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 21 2009: 
               (Start)
%C A068106 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.
%C A068106 Mirror image of A047920.
%C A068106 (End)
%C A068106 Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 18 2009: 
               (Start)
%C A068106 Sum[(k+1)*T(n,k), k=0..n]=A000166(n+2) (the derangement numbers).
%C A068106 (End)
%D A068106 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.
%D A068106 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]
%H A068106 D. Dumont, <a href="http://www.mat.univie.ac.at/~slc/opapers/s05dumont.html">
               Matrices d'Euler-Seidel</a>, Sem. Loth. Comb. B05c (1981) 59-78.
%F A068106 T(n, k) = Sum_{ j>= 0} (-1)^j*binomial(n-k, j)*(n-j)! . - Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), May 29 2005
%F A068106 Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 18 2009: 
               (Start)
%F A068106 T(n,k)=Sum[d(n-j)*binom(k, j), j=0..k], where d(i)=A000166(i) are the 
               derangement numbers.
%F A068106 (End)
%e A068106 Triangle begins
%e A068106 1..1..2..6..24..120..720...
%e A068106 ...0..1..4..18..96...600...
%e A068106 ......1..3..14..78...504...
%e A068106 .........2..11..64...426...
%e A068106 ............9...53...362...
%e A068106 ................44...309...
%e A068106 .....................265...
%p A068106 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]
%Y A068106 Row sums give A002467. Diagonals include A000166, A000255, A055790, A000142.
%Y A068106 See A047920 for another version.
%Y A068106 (When seen as array) Main diagonal is in A033815.
%Y A068106 A047920 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 21 2009]
%Y A068106 Sequence in context: A068598 A163770 A035561 this_sequence A005856 A157876 
               A107293
%Y A068106 Adjacent sequences: A068103 A068104 A068105 this_sequence A068107 A068108 
               A068109
%K A068106 nonn,easy,tabl,nice
%O A068106 0,6
%A A068106 N. J. A. Sloane (njas(AT)research.att.com), Apr 12 2002
%E A068106 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 
               01 2003