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

1, 1, 36, 291600, 1851776640000, 23813032808678400000000, 1333916640950593574375424000000000000, 618764594221522786972353235328676003840000000000000000

Offset

0,3

Comments

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.

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.

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 + …

The next term is too large to include.

Links

Ilya Gutkovskiy, Table of n, a(n) for n = 0..25

Ilya Gutkovskiy, Polynomials T_n(x)

Eric Weisstein's World of Mathematics, Triangular Number

Example

The first few polynomials are:
T_0(x) = x;
T_1(x) = x^2 - x;
T_2(x) = x^3 - 4*x^2 + 3*x;
T_3(x) = x^4 - 10*x^3 + 27*x^2 - 18*x;
T_4(x) = x^5 - 20*x^4 + 127*x^3 - 288*x^2 + 180*x;.
T_5(x) = x^6 - 35*x^5 + 427*x^4 - 2193*x^3 + 4500*x^2 - 2700*x, etc.
…
a(3) = discriminant T_3(x) = 291600.

Mathematica

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}]

Prog

(PARI) a(n) = poldisc(prod(k=0, n, 'x - k*(k + 1)/2)); \\ Michel Marcus, Mar 01 2023

Crossrefs

Cf. A000217, A128813.

Sequence in context: A334409 A216832 A013839 * A065752 A367543 A071226

Adjacent sequences: A271383 A271384 A271385 * A271387 A271388 A271389

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 06 2016

Status

approved