%I #6 Jul 20 2021 01:33:12

%S 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,

%T 3125,1,1,46656,37500,34560,34560,37500,46656,1,1,823543,5103,590625,

%U 35,590625,5103,823543,1,1,16777216,13176688,1792,11200000,11200000,1792,13176688,16777216,1

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

%H G. C. Greubel, <a href="/A128433/b128433.txt">Rows n = 0..50 of the triangle, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BernsteinPolynomial.html">Bernstein Polynomial</a>

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

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

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

%F T(n,0) = 1.

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

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 4, 4, 1;

%e 1, 27, 3, 27, 1;

%e 1, 256, 216, 216, 256, 1;

%e 1, 3125, 80, 5, 80, 3125, 1;

%e 1, 46656, 37500, 34560, 34560, 37500, 46656, 1;

%e 1, 823543, 5103, 590625, 35, 590625, 5103, 823543, 1;

%e 1, 16777216, 13176688, 1792, 11200000, 11200000, 1792, 13176688, 16777216, 1;

%t B[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k]*k^k*(n-k)^(n-k)/n^n];

%t T[n_, k_]= Numerator[B[n, k]];

%t Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 19 2021 *)

%o (Sage)

%o def B(n,k): return 1 if (k==0 or k==n) else binomial(n, k)*k^k*(n-k)^(n-k)/n^n

%o def T(n,k): return numerator(B(n,k))

%o flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Jul 19 2021

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

%K nonn,tabl,frac

%O 0,8

%A _Reinhard Zumkeller_, Mar 03 2007