A156600 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 = 6, read by rows.

1, 1, 1, 1, -5, 1, 1, 24, 24, 1, 1, -115, 552, -115, 1, 1, 551, 12673, 12673, 551, 1, 1, -2640, 290928, -1394030, 290928, -2640, 1, 1, 12649, 6678672, 153331178, 153331178, 6678672, 12649, 1, 1, -60605, 153318529, -16865038190, 80805530806, -16865038190, 153318529, -60605, 1

Offset

0,5

Links

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

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 = 6.

Example

Triangle begins as:
  1;
  1,     1;
  1,    -5,       1;
  1,    24,      24,         1;
  1,  -115,     552,      -115,         1;
  1,   551,   12673,     12673,       551,       1;
  1, -2640,  290928,  -1394030,    290928,   -2640,     1;
  1, 12649, 6678672, 153331178, 153331178, 6678672, 12649, 1;

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, 6], {n, 0, 12}, {k, 0, n}]//TableForm (* modified by G. C. Greubel, 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, 6], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 25 2021 *)

Prog

(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, 6) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 25 2021

Crossrefs

Cf. A007318 (m=0), A034801 (m=4), A156599 (m=5), this sequence (m=6), A156601 (m=7), A156602 (m=8), A156603.

Cf. A053122.

Sequence in context: A022168 A359993 A157212 * A329118 A152572 A203346

Adjacent sequences: A156597 A156598 A156599 * A156601 A156602 A156603

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 11 2009

Extensions

Edited by G. C. Greubel, Jun 25 2021

Status

approved