A094791 - OEIS

%I #33 Aug 14 2021 15:33:34

%S 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,

%T 689,265,1,21,196,1015,3094,5453,5053,1854,1,28,350,2492,10899,29596,

%U 48082,42048,14833,1,36,582,5460,32403,124908,309602,470328,391641,133496

%N Triangle read by rows giving coefficients of polynomials arising in successive differences of (n!)_{n>=0}.

%C 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.

%C 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). - _Olivier Gérard_, Jul 31 2011

%F T(n, n) = A000166(n).

%F T(2, k) = A000217(k).

%F Sum_{k=0..n} T(n,n-k)*x^k = Sum_{i=0..n} der(n,i)*binomial( n+x, i) (an analog of Worpitzky's identity). - _Olivier Gérard_, Jul 31 2011

%F The n-th row polynomial R(n,x) = Sum _{k = 0..n} T(n,k)*x^k is P-recursive in the variable x: x*R(n,x) = (x+n+1)*R(n,x-1) - R(n,x-2). - _Peter Bala_, Jul 25 2021

%e D_3(n) = n!*(n^3 + 3*n^2 + 5*n + 2).

%e D_4(n) = n!*(n^4 + 6*n^3 + 17*n^2 + 20*n + 9).

%e Table begins:

%e   1

%e   1  0

%e   1  1   1

%e   1  3   5   2

%e   1  6  17  20    9

%e   1 10  45 100  109   44

%e   1 15 100 355  694  689  265

%e   ...

%p with(LREtools): A094791_row := proc(n)

%p delta(x!,x,n); simplify(%/x!); seq(coeff(%,x,n-j),j=0..n) end:

%p seq(print(A094791_row(n)),n=0..9); # _Peter Luschny_, Jan 09 2015

%t d[0][n_] := n!; d[k_][n_] := d[k][n] = d[k - 1][n + 1] - d[k - 1][n] // FullSimplify;

%t row[k_] := d[k][n]/n! // FullSimplify // CoefficientList[#, n]& // Reverse;

%t Array[row, 10, 0] // Flatten (* _Jean-François Alcover_, Aug 02 2019 *)

%Y Successive differences of factorial numbers: A001563, A001564, A001565, A001688, A001689, A023043.

%Y Rencontres numbers A008290. Partial derangements A098825.

%Y Row sum is A000255. Signed version in A126353.

%Y Cf. A094792, A094793, A094794, A094795.

%K nonn,tabl

%O 0,8

%A _Benoit Cloitre_, Jun 11 2004

%E Edited and T(0,0) corrected according to the author's definition by _Olivier Gérard_, Jul 31 2011