

A271386


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


1




OFFSET

0,3


COMMENTS

Discriminants of the polynomials T_n(x) = ((1)^n*2^(n1)*cos(Pi*sqrt(8 x+1)/2)*Gamma(nsqrt(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}]


CROSSREFS

Cf. A000217, A128813.
Sequence in context: A185960 A216832 A013839 * A065752 A071226 A134369
Adjacent sequences: A271383 A271384 A271385 * A271387 A271388 A271389


KEYWORD

nonn


AUTHOR

Ilya Gutkovskiy, Apr 06 2016


STATUS

approved



