A120078 Coefficient triangle of numerator polynomials appearing in certain column o.g.f.s related to the H-atom spectrum.

1, 4, -3, 36, -27, -5, 144, -108, -20, -7, 3600, -2700, -500, -175, -81, 3600, -2700, -500, -175, -81, -44, 176400, -132300, -24500, -8575, -3969, -2156, -1300, 705600, -529200, -98000, -34300, -15876, -8624, -5200, -3375, 6350400, -4762800, -882000, -308700, -142884, -77616, -46800, -30375, -20825

Offset: 1

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• sign
• tabl

Comments

The row polynomials P(n,x) = Sum_{k=1..n-1} a(n,k)*x^k, n >= 1, appear in the numerator of the o.g.f. for column n of the triangle of rationals A120072(m,n)/A120073(m,n), m >= 2, n = 1..m-1. P(n,x) has degree n-1.

See the W. Lang link under A120072 for the precise form of the o.g.f.s: G(x,n) = -dilog(1-x) + x*P(n,4)/*(A(n)*(n^2)*(1-x)), with A(n) = [1, 1, 4, 9, 144, 100, 3600, 11025, 78400, 63504, ...] = conjectured to be A027451(n), n >= 1.

Examples

			For n=2 the o.g.f. of A120072(m,2)/A120073(m,2) (=[5/36, 3/16, 21/100, 2/9, ...]) is G(x,2) = -dilog(1-x) + x*P(2,x)/(1*4*(1-x)) = -dilog(1-x) + x*(4-3*x)/(4*(1-x)).
Triangle begins:
       1;
       4,      -3;
      36,     -27,     -5;
     144,    -108,    -20,    -7;
    3600,   -2700,   -500,  -175,   -81;
    3600,   -2700,   -500,  -175,   -81,   -44;
  176400, -132300, -24500, -8575, -3969, -2156, -1300;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened
• Wolfdieter Lang, First ten rows

Crossrefs

Row sums (unsigned) give A120079.

Signed row sums conjectured to coincide with A027451.

Cf. A027451, A051418, A120072, A120073, A120079.

Programs

Magma

f:= func< n | n eq 1 select 1 else 1/n^2 -1/(n-1)^2 >;
A120078:= func< n,k | (Lcm([1..n]))^2*f(k) >;
[A120078(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 26 2023

Mathematica

Table[(Apply[LCM, Range[n]])^2*If[k==1, 1, (1-2*k)/(k*(k-1))^2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Apr 26 2023 *)

SageMath

def f(k): return 1 if (k==1) else 1/k^2 - 1/(k-1)^2
def A120078(n,k): return (lcm(range(1, n+1)))^2*f(k)
flatten([[A120078(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Apr 26 2023

Formula

T(n, k) = A051418(n) * (1 if k = 1 otherwise 1/k^2 - 1/(k-1)^2). - G. C. Greubel, Apr 26 2023