A156609 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 = 3, read by rows.

1, 1, 1, 1, -2, 1, 1, 4, 4, 1, 1, -4, 8, -4, 1, 1, 4, 8, 8, 4, 1, 1, -4, 8, -8, 8, -4, 1, 1, 4, 8, 8, 8, 8, 4, 1, 1, -4, 8, -8, 8, -8, 8, -4, 1, 1, 4, 8, 8, 8, 8, 8, 8, 4, 1, 1, -4, 8, -8, 8, -8, 8, -8, 8, -4, 1

Offset: 0

Author

Roger L. Bagula, Feb 11 2009

Keywords

• sign
• tabl

Comments

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

Examples

			Triangle begins:
  1;
  1,  1;
  1, -2, 1;
  1,  4, 4,  1;
  1, -4, 8, -4, 1;
  1,  4, 8,  8, 4,  1;
  1, -4, 8, -8, 8, -4, 1;
  1,  4, 8,  8, 8,  8, 4,  1;
  1, -4, 8, -8, 8, -8, 8, -4, 1;
  1,  4, 8,  8, 8,  8, 8,  8, 4,  1;
  1, -4, 8, -8, 8, -8, 8, -8, 8, -4, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Crossrefs

Cf. A129862, A007318 (m=0), A156608 (m=2), this sequence (m=3), A156610 (m=4), A156612.

Programs

Magma

function T(n,k)
  if k eq 0 or k eq n then return 1;
  elif n eq 2 and k eq 1 then return -2;
  elif k eq 1 or k eq n-1 then return 4*(-1)^(n+1);
  elif k eq 2 or k eq n-2 then return 8;
  elif (n mod 2) eq 0 then return 8*(-1)^k;
  else return 8;
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 24 2021

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, 3], {n,0,15}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 23 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, 3], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 23 2021 *)

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,3) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jun 23 2021

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 = 3.

T(n, k) defined by T(n, 0) = T(n, n) = 1, T(2, 1) = -2, T(n, 1) = T(n, n-1) = 4*(-1)^(n+1), T(n, 2) = T(n, n-2) = 8, T(n, k) = 8*(-1)^k if n mod 2 = 0, and T(n, k) = 8 otherwise. - G. C. Greubel, Jun 24 2021

Extensions

Definition corrected and edited by G. C. Greubel, Jun 23 2021