A156599 Triangle T(n, k, m) = t(n, m)/(t(k, m)*t(n-k, m)), where 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 m = 5, read by rows.

1, 1, 1, 1, -4, 1, 1, 15, 15, 1, 1, -56, 210, -56, 1, 1, 209, 2926, 2926, 209, 1, 1, -780, 40755, -152152, 40755, -780, 1, 1, 2911, 567645, 7909187, 7909187, 567645, 2911, 1, 1, -10864, 7906276, -411126352, 1534382278, -411126352, 7906276, -10864, 1, 1, 40545, 110120220, 21370664028, 297662820390, 297662820390, 21370664028, 110120220, 40545, 1

Offset: 0

Author

Roger L. Bagula, Feb 11 2009

Keywords

• sign
• tabl
• less

Examples

			Triangle begins:
  1;
  1,    1;
  1,   -4,      1;
  1,   15,     15,       1;
  1,  -56,    210,     -56,       1;
  1,  209,   2926,    2926,     209,      1;
  1, -780,  40755, -152152,   40755,   -780,    1;
  1, 2911, 567645, 7909187, 7909187, 567645, 2911, 1;
		

Links

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

Crossrefs

Cf. A007318 (m=0), A034801 (m=4), this sequence (m=5), A156600 (m=6), A156601 (m=7), A156602 (m=8), A156603.

Cf. A053122.

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)];
T[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])];
Table[T[n, k, 5], {n,0,12}, {k,0,n}]//TableForm (* modified by G. C. Greubel, May 23 2019; Jun 25 2021 *)
(* Second program *)
t[n_, k_]:= t[n, k]= If[n==0, 1, If[k==0, (n-1)!, Product[(-1)^j*Simplify[ChebyshevU[j, x/2 - 1]], {j,0,n-1}]/.x->(k+1)]];
T[n_, k_, m_]:= T[n, k, m]= t[n, m]/(t[k, m]*t[n-k, m]);
Table[T[n, k, 5], {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-1)
    else: return product( (-1)^j*chebyshev_U(j, (k-1)/2) for j in (0..n-1) )
def T(n,k,m): return t(n,m)/(t(k,m)*t(n-k,m))
flatten([[T(n, k, 5) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 25 2021

Formula

T(n, k, m) = t(n, m)/(t(k, m)*t(n-k, m)), where 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 m = 5.

Extensions

Edited by G. C. Greubel, May 23 2019; Jun 25 2021