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

%I A128433 #7 Feb 16 2025 08:33:04
%S A128433 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 A128433 3125,1,1,46656,37500,34560,34560,37500,46656,1,1,823543,5103,590625,
%U A128433 35,590625,5103,823543,1,1,16777216,13176688,1792,11200000,11200000,1792,13176688,16777216,1
%N A128433 Triangle, read by rows, T(n,k) = numerator of the maximum of the k-th Bernstein polynomial of degree n; denominator is A128434.
%H A128433 G. C. Greubel, <a href="/A128433/b128433.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A128433 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BernsteinPolynomial.html">Bernstein Polynomial</a>
%F A128433 T(n,k)/A128434(n,k) = Binomial(n,k) * k^k * (n-k)^(n-k) / n^n.
%F A128433 For n>0: Sum_{k=0..n} T(n,k)/A128434(n,k) = A090878(n)/A036505(n-1).
%F A128433 T(n,n-k) = T(n,k).
%F A128433 T(n,0) = 1.
%F A128433 for n>0: T(n,1)/A128434(n,1) = A000312(n-1)/A000169(n).
%e A128433 Triangle begins as:
%e A128433   1;
%e A128433   1,        1;
%e A128433   1,        1,        1;
%e A128433   1,        4,        4,      1;
%e A128433   1,       27,        3,     27,        1;
%e A128433   1,      256,      216,    216,      256,        1;
%e A128433   1,     3125,       80,      5,       80,     3125,     1;
%e A128433   1,    46656,    37500,  34560,    34560,    37500, 46656,        1;
%e A128433   1,   823543,     5103, 590625,       35,   590625,  5103,   823543,        1;
%e A128433   1, 16777216, 13176688,   1792, 11200000, 11200000,  1792, 13176688, 16777216, 1;
%t A128433 B[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k]*k^k*(n-k)^(n-k)/n^n];
%t A128433 T[n_, k_]= Numerator[B[n, k]];
%t A128433 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 19 2021 *)
%o A128433 (Sage)
%o A128433 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 A128433 def T(n,k): return numerator(B(n,k))
%o A128433 flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Jul 19 2021
%Y A128433 Cf. A000169, A000312, A036505, A090878, A128434.
%K A128433 nonn,tabl,frac
%O A128433 0,8
%A A128433 _Reinhard Zumkeller_, Mar 03 2007