%I A047920
%S A047920 1,1,0,2,1,1,6,4,3,2,24,18,14,11,9,120,96,78,64,53,44,720,600,504,
%T A047920 426,362,309,265,5040,4320,3720,3216,2790,2428,2119,1854,40320,
%U A047920 35280,30960,27240,24024,21234,18806,16687,14833,362880,322560
%N A047920 Triangular array formed from successive differences of factorial numbers.
%C A047920 Number of permutations of 1,2,...,k,n+1,n+2,...,2n-k that have no agreements 
               with 1,...,n. For example consider 1234 and 1256, then n=4 and k=2, 
               so T(4,2)=14. Compare A000255 for the case k=1. - Jon Perry (perry(AT)globalnet.co.uk), 
               Jan 23 2004
%C A047920 Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 21 2009: 
               (Start)
%C A047920 T(n-1,k-1) is the number of non-derangements of {1,2,...,n} having smallest 
               fixed point equal to k. Example: T(3,1)=4 because we have 4213, 4231, 
               3214, and 3241 (the permutations of {1,2,3,4} having smallest fixed 
               equal to 2).
%C A047920 Row sums give the number of non-derangement permutations of {1,2,...,
               n} (A002467).
%C A047920 Mirror image of A068106.
%C A047920 Closely related to A134830, where each row has an extra term (see the 
               Charalambides reference).
%C A047920 (End)
%C A047920 Sum[(k+1)*T(n,k), k=0..n]=A155521(n+1). - [From Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Jul 18 2009]
%D A047920 J. D. H. Dickson, Discussion of two double series arising from the number 
               of terms in determinants of certain forms, Proc. London Math. Soc., 
               10 (1879), 120-122.
%D A047920 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]
%D A047920 Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, 
               Boca Raton, Florida, 2002, p. 176, Table 5.3. [From Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Apr 21 2009]
%H A047920 <a href="Sindx_Fa.html#factorial">Index entries for sequences related 
               to factorial numbers</a>
%F A047920 t(n, k) =t(n, k-1)-t(n-1, k-1) =t(n, k+1)-t(n-1, k) =n*t(n-1, k)+k*t(n-2, 
               k-1) =(n-1)*t(n-1, k-1)+(k-1)*t(n-2, k-2) =A060475(n, k)*(n-k)! - 
               Henry Bottomley (se16(AT)btinternet.com), Mar 16 2001
%F A047920 T(n, k) = Sum_{ j>= 0} (-1)^j * binomial(k, j)*(n-j)! . - Philippe DELEHAM, 
               May 29 2005
%F A047920 T(n,k)=Sum[d(n-j)*binom(n-k,j), j=0..n-k], where d(i)=A000166(i) are 
               the derangement numbers. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Jul 17 2009
%e A047920 1; 1,0; 2,1,1; 6,4,3,2; 24,18,14,11,9; 120,96,78,64,53,44; ...
%p A047920 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(n-k, j)*d[n-j], j = 0 .. n-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 17 2009]
%Y A047920 Columns give A000142, A001563, A001564, etc. Cf. A047922.
%Y A047920 See A068106 for another version of this triangle.
%Y A047920 Orthogonal columns: A000166, A000255, A055790. Main diagonal A033815.
%Y A047920 A068106, A002467, A134830 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Apr 21 2009]
%Y A047920 Cf. A155521.
%Y A047920 Sequence in context: A158389 A103880 A135899 this_sequence A144655 A140956 
               A166919
%Y A047920 Adjacent sequences: A047917 A047918 A047919 this_sequence A047921 A047922 
               A047923
%K A047920 nonn,tabl,easy,nice
%O A047920 0,4
%A A047920 N. J. A. Sloane (njas(AT)research.att.com).