A094791 Triangle read by rows giving coefficients of polynomials arising in successive differences of (n!)_{n>=0}.

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, 133496

Offset

0,8

Comments

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). - Olivier Gérard, Jul 31 2011

Links

Table of n, a(n) for n=0..54.

Formula

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

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

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

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

Example

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
  ...

Maple

with(LREtools): A094791_row := proc(n)
delta(x!, x, n); simplify(%/x!); seq(coeff(%, x, n-j), j=0..n) end:
seq(print(A094791_row(n)), n=0..9); # Peter Luschny, Jan 09 2015

Mathematica

d[0][n_] := n!; d[k_][n_] := d[k][n] = d[k - 1][n + 1] - d[k - 1][n] // FullSimplify;
row[k_] := d[k][n]/n! // FullSimplify // CoefficientList[#, n]& // Reverse;
Array[row, 10, 0] // Flatten (* Jean-François Alcover, Aug 02 2019 *)

Crossrefs

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.

Cf. A094792, A094793, A094794, A094795.

Sequence in context: A308180 A329633 A126353 * A243524 A349988 A272300

Adjacent sequences: A094788 A094789 A094790 * A094792 A094793 A094794

Keyword

nonn,tabl

Author

Benoit Cloitre, Jun 11 2004

Extensions

Edited and T(0,0) corrected according to the author's definition by Olivier Gérard, Jul 31 2011

Status

approved