A156603 Square array T(n, k) = Product_{j=1..n} p(j, k+1), p(n, x) = Sum_{j=0..n} (-1)^j*A053122(n, j)*x^j, and T(n, 0) = n!, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 0, 6, 1, 1, -1, 0, 24, 1, 1, -2, 0, 0, 120, 1, 1, -3, -6, 0, 0, 720, 1, 1, -4, -24, 24, 0, 0, 5040, 1, 1, -5, -60, 504, 120, 0, 0, 40320, 1, 1, -6, -120, 3360, 27720, -720, 0, 0, 362880, 1, 1, -7, -210, 13800, 702240, -3991680, -5040, 0, 0, 3628800

Offset: 0

Author

Roger L. Bagula, Feb 11 2009

Keywords

• sign
• tabl

Examples

			Square array begins as:
    1, 1,  1,    1,        1,          1,            1 ...;
    1, 1,  1,    1,        1,          1,            1 ...;
    2, 0, -1,  -2,        -3,         -4,           -5 ...;
    6, 0,  0,   -6,      -24,        -60,         -120 ...;
   24, 0,  0,   24,      504,       3360,        13800 ...;
  120, 0,  0,  120,    27720,     702240,      7603800 ...;
  720, 0,  0, -720, -3991680, -547747200, -20074032000 ...;
Antidiagonal rows begin as:
  1;
  1, 1;
  1, 1,  2;
  1, 1,  0,    6;
  1, 1, -1,    0,   24;
  1, 1, -2,    0,     0,    120;
  1, 1, -3,   -6,     0,      0,      720;
  1, 1, -4,  -24,    24,      0,        0,  5040;
  1, 1, -5,  -60,   504,    120,        0,     0, 40320;
  1, 1, -6, -120,  3360,  27720,     -720,     0,     0, 362880;
  1, 1, -7, -210, 13800, 702240, -3991680, -5040,     0,      0, 3628800;
		

Links

• G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Crossrefs

Cf. A007318, A034801, A053122, A156599, A156600, A156601, A156602.

Programs

Mathematica

(* First program *)
b[n_, k_]:= If[k==n, 2, If[k==n-1 || k==n+1, -1, 0]];
M[d_]:= Table[b[n, k], {n, d}, {k, d}];
p[x_, n_]:= If[n==0, 1, CharacteristicPolynomial[M[n], x]];
f = Table[p[x, n], {n, 0, 20}];
T[n_, k_]:= If[k==0, n!, Product[f[[j]], {j, n}]/.x->(k+1)];
Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 25 2021 *)
(* Second program *)
T[n_, k_]:= If[n==0, 1, If[k==0, n!, Product[(-1)^j*Simplify[ChebyshevU[j, x/2-1]], {j, 0, n-1}]/.x->(k+1)]];
Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 25 2021 *)

Sage

@CachedFunction
def T(n, k):
    if (n==0): return 1
    elif (k==0): return factorial(n)
    else: return product( (-1)^j*chebyshev_U(j, (k-1)/2) for j in (0..n-1) )
flatten([[T(k, n-k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jun 25 2021

Formula

T(n, k) = Product_{j=1..n} p(j, k+1), p(n, x) = Sum_{j=0..n} (-1)^j*A053122(n, j)*x^j, and T(n, 0) = n! (square array).

T(n, k) = Product_{j=0..n-1} (-1)^j*ChebyshevU(j, (k-1)/2) with T(n, 0) = n! for n >= 1, and T(0, k) = 1 (square array). - G. C. Greubel, Jun 25 2021

Extensions

Edited by G. C. Greubel, Jun 25 2021