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

0,5

Links

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

Eric Weisstein's World of Mathematics, Bernstein Polynomial

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).

Example

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;

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 *)

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 denominator(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, A128433.

Sequence in context: A229962 A141601 A108558 * A176417 A119731 A368928

Adjacent sequences: A128431 A128432 A128433 * A128435 A128436 A128437

Keyword

nonn,tabl,frac

Author

Reinhard Zumkeller, Mar 03 2007

Status

approved