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

%I A128434 #8 Feb 16 2025 08:33:04
%S A128434 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,
%T A128434 243,7776,1,1,117649,117649,117649,117649,117649,117649,1,1,2097152,
%U A128434 16384,2097152,128,2097152,16384,2097152,1,1,43046721,43046721,6561,43046721,43046721,6561,43046721,43046721,1
%N A128434 Triangle, read by rows, T(n,k) = denominator of the maximum of the k-th Bernstein polynomial of degree n; numerator is A128433.
%H A128434 G. C. Greubel, <a href="/A128434/b128434.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A128434 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BernsteinPolynomial.html">Bernstein Polynomial</a>
%F A128434 A128433(n,k)/T(n,k) = binomial(n,k) * k^k * (n-k)^(n-k) / n^n.
%F A128434 For n>0: Sum_{k=0..n} A128433(n,k)/T(n,k) = A090878(n)/A036505(n-1);
%F A128434 T(n, n-k) = T(n,k).
%F A128434 T(n, 0) = T(n, n) = 1.
%F A128434 for n>0: A128433(n,1)/T(n,1) = A000312(n-1)/A000169(n).
%e A128434 Triangle begins as:
%e A128434   1;
%e A128434   1,       1;
%e A128434   1,       2,      1;
%e A128434   1,       9,      9,       1;
%e A128434   1,      64,      8,      64,      1;
%e A128434   1,     625,    625,     625,    625,       1;
%e A128434   1,    7776     243,      16,    243,    7776,      1;
%e A128434   1,  117649, 117649,  117649, 117649,  117649, 117649,       1;
%e A128434   1, 2097152,  16384, 2097152,    128, 2097152,  16384, 2097152, 1;
%t A128434 B[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k]*k^k*(n-k)^(n-k)/n^n];
%t A128434 T[n_, k_]= Denominator[B[n, k]];
%t A128434 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 19 2021 *)
%o A128434 (Sage)
%o A128434 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 A128434 def T(n,k): return denominator(B(n,k))
%o A128434 flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Jul 19 2021
%Y A128434 Cf. A000169, A000312, A036505, A090878, A128433.
%K A128434 nonn,tabl,frac
%O A128434 0,5
%A A128434 _Reinhard Zumkeller_, Mar 03 2007