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
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
Programs
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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
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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
Comments