A172375 Triangle T(n, k, q) = c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k, q)), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = q*(f(n-1, q) + f(n-2, q)), f(0, q) = 0, f(1, q) = 1, and q = 2, read by rows.

1, 1, 1, 1, 6, 1, 1, 48, 48, 1, 1, 352, 2816, 352, 1, 1, 2640, 154880, 154880, 2640, 1, 1, 19680, 8659200, 63500800, 8659200, 19680, 1, 1, 146944, 481976320, 26508697600, 26508697600, 481976320, 146944, 1, 1, 1096704, 26859012096, 11012194959360, 82591462195200, 11012194959360, 26859012096, 1096704, 1

Offset

1,5

Links

G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Formula

T(n, k, q) = c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k, q)), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = q*(f(n-1, q) + f(n-2, q)), f(0, q) = 0, f(1, q) = 1, and q = 2.

T(n, k, q) = c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k,q)), where c(n, q) = Product_{j=1..n} f(j, q), and f(n, q) = (-I*sqrt(q))^(n-1)*ChebyshevU(n-1, i*sqrt(q)/2). - G. C. Greubel, May 07 2021

Example

Triangle begins as:
  1;
  1,      1;
  1,      6,         1;
  1,     48,        48,           1;
  1,    352,      2816,         352,           1;
  1,   2640,    154880,      154880,        2640,         1;
  1,  19680,   8659200,    63500800,     8659200,     19680,      1;
  1, 146944, 481976320, 26508697600, 26508697600, 481976320, 146944, 1;

Mathematica

f[n_, q_]:= (-I*Sqrt[q])^(n-1)*ChebyshevU[n-1, I*Sqrt[q]/2];
c[n_, q_]:= Product[f[j, q], {j, n}];
T[n_, k_, q_]:= c[n-1, q]*c[n, q]/(c[k-1, q]^2*c[n-k, q]*c[n-k+1, q]*f[k, q]);
Table[T[n, k, 2], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, May 07 2021 *)

Prog

(Sage)
@CachedFunction
def f(n, q): return (-i*sqrt(q))^(n-1)*chebyshev_U(n-1, i*sqrt(q)/2)
def c(n, q): return product( f(j, q) for j in (1..n) )
def T(n, k, q): return c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k, q))
flatten([[T(n, k, 2) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, May 07 2021

Crossrefs

Cf. A002605, A030195, this sequence (q=2), A172376 (q=3).

Sequence in context: A156765 A015117 A287020 * A075377 A046792 A209330

Adjacent sequences: A172372 A172373 A172374 * A172376 A172377 A172378

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 01 2010

Extensions

Edited by G. C. Greubel, May 07 2021

Status

approved