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A094791
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Triangle read by rows giving coefficients of polynomials arising in successive differences of (n!)_{n>=0}.
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1
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1, 1, 0, 1, 1, 1, 1, 3, 5, 2, 1, 6, 17, 20, 9, 1, 10, 45, 100, 109, 44, 1, 15, 100, 355, 694, 689, 265, 1, 21, 196, 1015, 3094, 5453, 5053, 1854, 1, 28, 350, 2492, 10899, 29596, 48082, 42048, 14833, 1, 36, 582, 5460, 32403, 124908, 309602, 470328, 391641
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OFFSET
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0,8
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COMMENTS
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Let D_0(n)=n! and D_{k+1}(n)=D_{k}(n+1)-D_{k}(n), then D_{k}(n)=n!*P_{k}(n) where P_{k} is a polynomial with integer coefficients of degree k.
The horizontal reversal of this triangle arises as a binomial convolution of the derangements coefficients der(n,i) (numbers of permutations of size n with i derangements = A098825(n,i) = number of permutations of size n with n-i rencontres = A008290(n,n-i), see formula section) [From Olivier Gerard, Jul 31 2011]
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LINKS
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Table of n, a(n) for n=0..53.
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FORMULA
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T(n, n)=A000166; T(2, k)=A000217
sum( T(n,n-k) x^k, k=0..n) = sum( der(n,i)*binomial( n+x, i), i=0..n) (an analogue of Worpitzky's identity) [From Olivier Gerard, Jul 31 2011]
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EXAMPLE
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D_3(n)=n!*(n^3 + 3*n^2 + 5*n + 2); D_4(n)=n!*(n^4 + 6*n^3 + 17*n^2 + 20*n + 9)
Table begins
1
1 0
1 1 1
1 3 5 2
1 6 17 20 9
1 10 45 100 109 44
1 15 100 355 694 689 265
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CROSSREFS
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Successive differences of factorial numbers: A001563, A001564, A001565, A001688, A001689, A023043
Rencontres numbers A008290.
Partial derangements A098825.
Row sum is A000255. Signed version in A126353.
Sequence in context: A161865 A145325 A126353 * A115406 A059246 A091276
Adjacent sequences: A094788 A094789 A094790 * A094792 A094793 A094794
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KEYWORD
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nonn,tabl
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AUTHOR
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Benoit Cloitre, Jun 11 2004
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EXTENSIONS
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Edited and T(0,0) corrected according to the author's definition by Olivier Gérard, Jul 31 2011
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STATUS
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approved
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