A156610 Triangle T(n, k, m) = round( t(n,m)/(t(k,m)*t(n-k,m)) ), with T(0, k, m) = 1, where t(n, k) = Product_{j=1..n} A129862(k+1, j), t(n, 0) = n!, and m = 4, read by rows.

1, 1, 1, 1, -3, 1, 1, 9, 9, 1, 1, -21, 63, -21, 1, 1, 54, 378, 378, 54, 1, 1, -141, 2538, -5922, 2538, -141, 1, 1, 369, 17343, 104058, 104058, 17343, 369, 1, 1, -966, 118818, -1861482, 4786668, -1861482, 118818, -966, 1, 1, 2529, 814338, 33387858, 224175618, 224175618, 33387858, 814338, 2529, 1

Offset

0,5

Comments

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

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k, m) = round( t(n,m)/(t(k,m)*t(n-k,m)) ), with T(0, k, m) = 1, where t(n, k) = Product_{j=1..n} A129862(k+1, j), t(n, 0) = n!, and m = 4.

T(n, 1) = T(n, n-1) = [n==1] - 3*A219233(n-2)*[n >= 2]. - G. C. Greubel, Jun 24 2021

Example

Triangle begins:
  1;
  1,    1;
  1,   -3,      1;
  1,    9,      9,        1;
  1,  -21,     63,      -21,         1;
  1,   54,    378,      378,        54,         1;
  1, -141,   2538,    -5922,      2538,      -141,        1;
  1,  369,  17343,   104058,    104058,     17343,      369,      1;
  1, -966, 118818, -1861482,   4786668,  -1861482,   118818,   -966,    1;
  1, 2529, 814338, 33387858, 224175618, 224175618, 33387858, 814338, 2529, 1;

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, 20}];
t[n_, k_]:= If[k==0, n!, Product[f[[j+1]], {j, n-1}]]/.x -> k+1;
T[n_, k_, m_]:= Round[t[n, m]/(t[k, m]*t[n-k, m])];
Table[T[n, k, 4], {n, 0, 15}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 24 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);
T[n_, k_, m_]:= T[n, k, m]= Round[t[n, m]/(t[k, m]*t[n-k, m])];
Table[T[n, k, 4], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 24 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 g(n, k): return factorial(n) if (k==0) else product( f(j, k+1) for j in (1..n-1) )
def T(n, k, m): return round( g(n, m)/(g(k, m)*g(n-k, m)) )
flatten([[T(n, k, 4) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jun 24 2021

Crossrefs

Cf. A129862, A007318 (m=0), A156608 (m=2), A156609 (m=3), this sequence (m=4), A156612.

Sequence in context: A204180 A319729 A106340 * A203460 A157179 A152655

Adjacent sequences: A156607 A156608 A156609 * A156611 A156612 A156613

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 11 2009

Extensions

Definition corrected and edited by G. C. Greubel, Jun 24 2021

Status

approved