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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.
2
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
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
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 01 2010
EXTENSIONS
Edited by G. C. Greubel, May 07 2021
STATUS
approved