A128813 -id:A128813 - cp's OEIS frontend

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

Ilya Gutkovskiy, Apr 06 2016

nonn

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.

			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.

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

Cf. A000217, A128813.

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}]

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

A108959 Triangle arising in connection with deformations of type D Kleinian singularities.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 10, 14, 7, 5, 20, 54, 76, 38, 6, 35, 154, 419, 590, 295, 7, 56, 364, 1616, 4400, 6196, 3098, 8, 84, 756, 4962, 22048, 60036, 84542, 42271, 9, 120, 1428, 12984, 85300, 379052, 1032154, 1453468, 726734, 10, 165, 2508, 30162, 274516, 1803638, 8014990, 21824737, 30733358, 15366679

Offset: 0

Paul Boddington, Jul 22 2005

			Triangle begins:
  1;
  2,  1;
  3,  4,  2;
  4, 10, 14,  7;
  5, 20, 54, 76, 38;
  ...

Paul Boddington, No-cycle algebras and representation theory, Ph.D. thesis, University of Warwick, 2004.

This sequence is an improved version of A097418. Coefficients of 1 give A000366.

Cf. A128813 (the p_k polynomials).

tabl(nn) = my(v = vector(nn)); for (n=1, nn, my(p=prod(i=1, n, x+i*(i-1)/2), q=n*p/x); v[n] = q - sum(i=1, n-1, polcoeff(p, i)*v[i])); vector(nn, k, Vec(v[k])); \\ Michel Marcus, Mar 18 2023

For k>=0 define p_k(x) = x(x+1)(x+3)...(x+k(k-1)/2) and consider the linear map taking each p_k(x) to k*p_k(x)/x. Then the images of x, x^2, x^3, ... are given by the rows. E.g., x^3 goes to 3x^2 + 4x + 2.

More terms from Michel Marcus, Mar 18 2023

cp's OEIS Frontend

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

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