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

1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 27, 3, 27, 1, 1, 256, 216, 216, 256, 1, 1, 3125, 80, 5, 80, 3125, 1, 1, 46656, 37500, 34560, 34560, 37500, 46656, 1, 1, 823543, 5103, 590625, 35, 590625, 5103, 823543, 1, 1, 16777216, 13176688, 1792, 11200000, 11200000, 1792, 13176688, 16777216, 1

Offset: 0

Author

Reinhard Zumkeller, Mar 03 2007

Keywords

• nonn
• tabl
• frac

Examples

			Triangle begins as:
  1;
  1,        1;
  1,        1,        1;
  1,        4,        4,      1;
  1,       27,        3,     27,        1;
  1,      256,      216,    216,      256,        1;
  1,     3125,       80,      5,       80,     3125,     1;
  1,    46656,    37500,  34560,    34560,    37500, 46656,        1;
  1,   823543,     5103, 590625,       35,   590625,  5103,   823543,        1;
  1, 16777216, 13176688,   1792, 11200000, 11200000,  1792, 13176688, 16777216, 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, A128434.

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_]= Numerator[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 numerator(B(n,k))
flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jul 19 2021

Formula

T(n,k)/A128434(n,k) = Binomial(n,k) * k^k * (n-k)^(n-k) / n^n.

For n>0: Sum_{k=0..n} T(n,k)/A128434(n,k) = A090878(n)/A036505(n-1).

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

T(n,0) = 1.

for n>0: T(n,1)/A128434(n,1) = A000312(n-1)/A000169(n).