VOOZH about

URL: https://oeis.org/A128433

⇱ A128433 - OEIS

login
A128433
Triangle, read by rows, T(n,k) = numerator of the maximum of the k-th Bernstein polynomial of degree n; denominator is A128434.
11
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
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
(SageMath)
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
KEYWORD
nonn,tabl,frac
AUTHOR
Reinhard Zumkeller, Mar 03 2007
STATUS
approved