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A271386 Discriminants of the polynomials T_n(x) = Product_{k=0..n} (x - k*(k + 1)/2).

%I A271386 #17 Feb 16 2025 08:33:33
%S A271386 1,1,36,291600,1851776640000,23813032808678400000000,
%T A271386 1333916640950593574375424000000000000,
%U A271386 618764594221522786972353235328676003840000000000000000
%N A271386 Discriminants of the polynomials T_n(x) = Product_{k=0..n} (x - k*(k + 1)/2).
%C A271386 Discriminants of the polynomials T_n(x) = -((-1)^n*2^(-n-1)*cos(Pi*sqrt(8 x+1)/2)*Gamma(n-sqrt(8 x+1)/2+3/2)*Gamma(n+sqrt(8 x+1)/2+3/2))/Pi, where Gamma(x) is the gamma function.
%C A271386 T_n (x) is described as a polynomial of degree (n + 1) with leading coefficient 1, and with first (n+1) triangular numbers as roots.
%C A271386 T_n(x) have generating function G(x,t) = x +  (x^2 - x)*t +  (x^3 - 4*x^2 + 3*x)*t^2 + (x^4 - 10*x^3 + 27*x^2 - 18*x)*t^3 + …
%C A271386 The next term is too large to include.
%H A271386 Ilya Gutkovskiy, <a href="/A271386/b271386.txt">Table of n, a(n) for n = 0..25</a>
%H A271386 Ilya Gutkovskiy, <a href="/A271386/a271386.pdf">Polynomials T_n(x)</a>
%H A271386 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>
%e A271386 The first few polynomials are:
%e A271386 T_0(x) = x;
%e A271386 T_1(x) = x^2 - x;
%e A271386 T_2(x) = x^3 - 4*x^2 + 3*x;
%e A271386 T_3(x) = x^4 - 10*x^3 + 27*x^2 - 18*x;
%e A271386 T_4(x) = x^5 - 20*x^4 + 127*x^3 - 288*x^2 + 180*x;.
%e A271386 T_5(x) = x^6 - 35*x^5 + 427*x^4 - 2193*x^3 + 4500*x^2 - 2700*x, etc.
%e A271386 …
%e A271386 a(3) = discriminant T_3(x) = 291600.
%t A271386 Table[Discriminant[(-1/2)^n x Pochhammer[3/2 - Sqrt[1 + 8 x]/2, n] Pochhammer[(3 + Sqrt[1 + 8 x])/2, n], x], {n, 0, 7}]
%o A271386 (PARI) a(n) = poldisc(prod(k=0, n, 'x - k*(k + 1)/2)); \\ _Michel Marcus_, Mar 01 2023
%Y A271386 Cf. A000217, A128813.
%K A271386 nonn
%O A271386 0,3
%A A271386 _Ilya Gutkovskiy_, Apr 06 2016