A156608 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 = 2, read by rows.

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

Offset: 0

Author

Roger L. Bagula, Feb 11 2009

Keywords

• sign
• tabl

Comments

The original definition of this sequence said it was based on the Cartan matrix of type D_n, so that matrix is somehow implicitly involved. - N. J. A. Sloane, Jun 25 2021

Examples

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

Links

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

Crossrefs

Cf. A129862, A007318 (m=0), this sequence (m=2), A156609 (m=3), A156610 (m=4), A156612.

Programs

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, 2], {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, 2], {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,2) 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 = 2.

Extensions

Edited by G. C. Greubel, Jun 23 2021