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

0,8

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Eric Weisstein's World of Mathematics, Bernstein Polynomial

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).

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A000169, A000312, A036505, A090878, A128434.

Sequence in context: A189150 A155194 A080044 * A089746 A225330 A094884

Adjacent sequences: A128430 A128431 A128432 * A128434 A128435 A128436

Keyword

nonn,tabl,frac

Author

Reinhard Zumkeller, Mar 03 2007

Status

approved