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A173585 Triangle T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)), where c(n, q) = Product_{j=1..n} t(2*j, q), t(n, q) = (1/4)*( (2 + sqrt(q))^n + (2 - sqrt(q))^n - 2 ), and q = 3, read by rows. 3
1, 1, 1, 1, 16, 1, 1, 225, 225, 1, 1, 3136, 44100, 3136, 1, 1, 43681, 8561476, 8561476, 43681, 1, 1, 608400, 1660970025, 23150231104, 1660970025, 608400, 1, 1, 8473921, 322220846025, 62555239000969, 62555239000969, 322220846025, 8473921, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
FORMULA
T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)), where c(n, q) = Product_{j=1..n} t(2*j, q), t(n, q) = (1/4)*( (2 + sqrt(q))^n + (2 - sqrt(q))^n - 2 ), and q = 3.
From G. C. Greubel, Jul 06 2021: (Start)
T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = (1/2^n)*Product_{j=1..n} (1 - ChebyshevT(2*j, k+1)), b(n, 0) = n!, and m = 1.
T(n, k, m) = Product_{j=1..n-k} ( (1 - ChebyshevT(2*j+2*k, m+1))/(1 - ChebyshevT(2*j, m+1)) ) with m = 1. (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 16, 1;
1, 225, 225, 1;
1, 3136, 44100, 3136, 1;
1, 43681, 8561476, 8561476, 43681, 1;
1, 608400, 1660970025, 23150231104, 1660970025, 608400, 1;
MATHEMATICA
(* First program *)
f[n_, q_]:= (1/4)*((2+Sqrt[q])^n + (2-Sqrt[q])^n -2);
c[n_, q_]:= Product[f[k, q], {k, 2, n, 2}]//Simplify;
T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n - k, q]);
Table[T[n, k, 3], {n, 0, 10, 2}, {k, 0, n, 2}]//Flatten (* modified by G. C. Greubel, Jul 06 2021 *)
(* Second program *)
t[n_, q_]:= (1/4)*(Round[(2+Sqrt[q])^n + (2-Sqrt[q])^n] -2);
c[n_, q_]:= Product[t[2*j, q], {j, n}];
T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);
Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 06 2021 *)
PROG
(Magma)
b:= func< n, k | n eq 0 select 1 else k eq 0 select Factorial(n) else (&*[1 - Evaluate(ChebyshevT(j), k+1)^2 : j in [1..n]]) >;
T:= func< n, k, m | b(n, m)/(b(k, m)*b(n-k, m)) >;
[T(n, k, 1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 06 2021
(Sage)
@CachedFunction
def f(n, q): return (1/4)*( round((2 + sqrt(q))^n + (2 - sqrt(q))^n) - 2 )
def c(n, q): return product( f(2*j, q) for j in (1..n))
def T(n, k, q): return c(n, q)/(c(k, q)*c(n-k, q))
flatten([[T(n, k, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 06 2021
CROSSREFS
Cf. A022168 (q=0), A022173 (q=1), this sequence (q=3).
Cf. A007318 (m=0), this sequence (m=1), A156645 (m=2), A156646 (m=10).
Sequence in context: A203397 A338029 A173885 * A022179 A015141 A176390
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 22 2010
EXTENSIONS
Edited by G. C. Greubel, Jul 06 2021
STATUS
approved

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Last modified April 23 13:41 EDT 2024. Contains 371914 sequences. (Running on oeis4.)