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A047920
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Triangular array formed from successive differences of factorial numbers.
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15
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1, 1, 0, 2, 1, 1, 6, 4, 3, 2, 24, 18, 14, 11, 9, 120, 96, 78, 64, 53, 44, 720, 600, 504, 426, 362, 309, 265, 5040, 4320, 3720, 3216, 2790, 2428, 2119, 1854, 40320, 35280, 30960, 27240, 24024, 21234, 18806, 16687, 14833, 362880, 322560
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OFFSET
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0,4
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COMMENTS
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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, Jan 23 2004
Contribution from Emeric Deutsch, Apr 21 2009: (Start)
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).
Row sums give the number of non-derangement permutations of {1,2,...,n} (A002467).
Mirror image of A068106.
Closely related to A134830, where each row has an extra term (see the Charalambides reference).
(End)
Sum[(k+1)*T(n,k), k=0..n]=A155521(n+1). - [From Emeric Deutsch, Jul 18 2009]
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REFERENCES
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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.
E. Deutsch and S. Elizalde, The largest and the smallest fixed points of permutations, arXiv:0904.2792v1, 2009. [From Emeric Deutsch, Apr 21 2009]
Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 176, Table 5.3. [From Emeric Deutsch, Apr 21 2009]
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LINKS
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_Reinhard Zumkeller_, Rows n=0..150 of triangle, flattened
Index entries for sequences related to factorial numbers
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FORMULA
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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, Mar 16 2001
T(n, k) = Sum_{ j>= 0} (-1)^j * binomial(k, j)*(n-j)! . - Philippe DELEHAM, May 29 2005
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, Jul 17 2009
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EXAMPLE
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1; 1,0; 2,1,1; 6,4,3,2; 24,18,14,11,9; 120,96,78,64,53,44; ...
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MAPLE
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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, Jul 17 2009]
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MATHEMATICA
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t[n_, k_] = Sum[(-1)^j*Binomial[k, j]*(n-j)!, {j, 0, n}]; Flatten[Table[t[n, k], {n, 0, 9}, {k, 0, n}]][[1 ;; 47]] (* From Jean-François Alcover, May 17 2011, after Ph. Deleham *)
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PROG
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(Haskell)
a047920 n k = a047920_tabl !! n !! k
a047920_row n = a047920_tabl !! n
a047920_tabl = map fst $ iterate e ([1], 1) where
e (row, n) = (scanl (-) (n * head row) row, n + 1)
-- Reinhard Zumkeller, Mar 05 2012
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CROSSREFS
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Columns give A000142, A001563, A001564, etc. Cf. A047922.
See A068106 for another version of this triangle.
Orthogonal columns: A000166, A000255, A055790. Main diagonal A033815.
A068106, A002467, A134830 [From Emeric Deutsch, Apr 21 2009]
Cf. A155521.
Sequence in context: A186023 A103880 A135899 * A144655 A190782 A199063
Adjacent sequences: A047917 A047918 A047919 * A047921 A047922 A047923
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KEYWORD
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nonn,tabl,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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