A128434 Triangle, read by rows, T(n,k) = denominator of the maximum of the k-th Bernstein polynomial of degree n; numerator is A128433.

1, 1, 1, 1, 2, 1, 1, 9, 9, 1, 1, 64, 8, 64, 1, 1, 625, 625, 625, 625, 1, 1, 7776, 243, 16, 243, 7776, 1, 1, 117649, 117649, 117649, 117649, 117649, 117649, 1, 1, 2097152, 16384, 2097152, 128, 2097152, 16384, 2097152, 1, 1, 43046721, 43046721, 6561, 43046721, 43046721, 6561, 43046721, 43046721, 1

Offset: 0

Author

Reinhard Zumkeller, Mar 03 2007

Keywords

• nonn
• tabl
• frac

Examples

			Triangle begins as:
  1;
  1,       1;
  1,       2,      1;
  1,       9,      9,       1;
  1,      64,      8,      64,      1;
  1,     625,    625,     625,    625,       1;
  1,    7776     243,      16,    243,    7776,      1;
  1,  117649, 117649,  117649, 117649,  117649, 117649,       1;
  1, 2097152,  16384, 2097152,    128, 2097152,  16384, 2097152, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• Eric Weisstein's World of Mathematics, Bernstein Polynomial

Crossrefs

Cf. A000169, A000312, A036505, A090878, A128433.

Programs

Mathematica

B[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k]*k^k*(n-k)^(n-k)/n^n];
T[n_, k_]= Denominator[B[n, k]];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 19 2021 *)

Sage

def B(n,k): return 1 if (k==0 or k==n) else binomial(n, k)*k^k*(n-k)^(n-k)/n^n
def T(n,k): return denominator(B(n,k))
flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jul 19 2021

Formula

A128433(n,k)/T(n,k) = binomial(n,k) * k^k * (n-k)^(n-k) / n^n.

For n>0: Sum_{k=0..n} A128433(n,k)/T(n,k) = A090878(n)/A036505(n-1);

T(n, n-k) = T(n,k).

T(n, 0) = T(n, n) = 1.

for n>0: A128433(n,1)/T(n,1) = A000312(n-1)/A000169(n).