A156612 Square array T(n, k) = Product_{j=1..n} A129862(k+1, j) with 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, -1, 0, 120, 1, 1, -3, -8, -1, 0, 720, 1, 1, -4, -27, 32, 2, 0, 5040, 1, 1, -5, -64, 567, 128, 2, 0, 40320, 1, 1, -6, -125, 3584, 30618, -512, 2, 0, 362880, 1, 1, -7, -216, 14375, 745472, -4317138, -2048, -4, 0, 3628800

Offset

0,6

Comments

Cartan_Dn refers to a Cartan matrix of type D_n. - N. J. A. Sloane, Jun 25 2021

Links

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

Formula

T(n, k) = Product_{j=1..n} A129862(k+1, j) with T(n, 0) = n!.

Example

Square array begins:
    1, 1,  1,   1,     1,      1 ...;
    1, 1,  1,   1,     1,      1 ...;
    2, 0, -1,  -2,    -3,     -4 ...;
    6, 0, -1,  -8,   -27,    -64 ...;
   24, 0, -1,  32,   567,   3584 ...;
  120, 0,  2, 128, 30618, 745472 ...;
Triangle begins as:
  1;
  1, 1;
  1, 1,  2;
  1, 1,  0,    6;
  1, 1, -1,    0,    24;
  1, 1, -2,   -1,     0,    120;
  1, 1, -3,   -8,    -1,      0,      720;
  1, 1, -4,  -27,    32,      2,        0,  5040;
  1, 1, -5,  -64,   567,    128,        2,     0, 40320;
  1, 1, -6, -125,  3584,  30618,     -512,     2,     0, 362880;
  1, 1, -7, -216, 14375, 745472, -4317138, -2048,    -4,      0, 3628800;

Mathematica

(* First program *)
b[n_, k_, d_]:= If[n==k, 2, If[(k==d && n==d-2) || (n==d && k==d-2), -1, If[(k==n- 1 || k==n+1) && n<=d-1 && k<=d-1, -1, 0]]];
M[d_]:= Table[b[n, k, d], {n, d}, {k, d}];
p[x_, n_]:= If[n==0, 1, CharacteristicPolynomial[M[n], x]];
f = Table[p[x, n], {n, 0, 30}];
T[n_, k_]:= If[k==0, n!, Product[f[[j+1]], {j, n-1}]]/.x -> k+1;
Table[T[k, n - k], {n, 0, 15}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 25 2021 *)
(* Second program *)
f[n_, x_]:= f[n, x]= If[n<2, (2-x)^n, (2-x)*LucasL[2*(n-1), Sqrt[-x]]];
t[n_, k_]:= t[n, k]= If[k==0, n!, Product[f[j, x], {j, n-1}]]/.x -> (k+1);
Table[t[k, n-k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 25 2021 *)

Prog

(Sage)
@CachedFunction
def f(n, x): return (2-x)^n if (n<2) else 2*(2-x)*sum( ((n-1)/(2*n-j-2))*binomial(2*n-j-2, j)*(-x)^(n-j-1) for j in (0..n-1) )
def T(n, k): return factorial(n) if (k==0) else product( f(j, k+1) for j in (1..n-1) )
flatten([[T(k, n-k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jun 25 2021

Crossrefs

Cf. A129862, A156608, A156609, A156610.

Sequence in context: A173591 A343320 A156603 * A351791 A368028 A342645

Adjacent sequences: A156609 A156610 A156611 * A156613 A156614 A156615

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 11 2009

Extensions

Edited by G. C. Greubel, Jun 25 2021

Status

approved