cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A156646 Triangle T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2), b(n, 0) = n!, and m = 10, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 484, 1, 1, 233289, 233289, 1, 1, 112444816, 54198633636, 112444816, 1, 1, 54198168025, 12591535188240100, 12591535188240100, 54198168025, 1, 1, 26123404543236, 2925290638056514680225, 1409984043580226203632400, 2925290638056514680225, 26123404543236, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 12 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,           1;
  1,         484,                 1;
  1,      233289,            233289,                 1;
  1,   112444816,       54198633636,         112444816,           1;
  1, 54198168025, 12591535188240100, 12591535188240100, 54198168025, 1;

Links

Crossrefs

Cf. A007318 (m=0), A173585 (m=1), A156645 (m=2), this sequence (m=10).
Cf. A123583, A156647.

Programs

  • 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,10): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 03 2021
  • Mathematica
    (* First program *)
b[n_, k_]:= b[n,k]= If[k==0, n!, Product[1 -ChebyshevT[j, k+1]^2, {j,n}]];
T[n_, k_, m_]= b[n,m]/(b[k,m]*b[n-k,m]);
Table[T[n,k,10], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jul 03 2021 *)
(* Second program *)
T[n_, k_, m_]:= T[n, k, m]= Product[(1 - ChebyshevT[2*j+2*k, m+1])/(1 - ChebyshevT[2*j, m+1]), {j, n-k}];
Table[T[n,k,12], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 03 2021 *)
  • Sage
    def b(n, k): return factorial(n) if (k==0) else product( 1 - chebyshev_T(j, k+1)^2 for j in (1..n) )
def T(n, k, m): return b(n,m)/(b(k,m)*b(n-k,m))
flatten([[T(n, k, 10) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 03 2021

Formula

T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2), b(n, 0) = n!, and m = 10.
From G. C. Greubel, Jul 03 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 = 10.
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 = 10. (End)

Extensions

Edited by G. C. Greubel, Jul 03 2021

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.

Original entry on oeis.org

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
Offset: 0

Views

Author

Roger L. Bagula, Feb 22 2010

Keywords

Examples

			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;

Links

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).
Cf. A123583, A156647.

Programs

  • 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
  • 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 *)
  • 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

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)

Extensions

Edited by G. C. Greubel, Jul 06 2021
Showing 1-2 of 2 results.

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