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

%I A120078 #23 May 03 2023 09:13:01
%S A120078 1,4,-3,36,-27,-5,144,-108,-20,-7,3600,-2700,-500,-175,-81,3600,-2700,
%T A120078 -500,-175,-81,-44,176400,-132300,-24500,-8575,-3969,-2156,-1300,
%U A120078 705600,-529200,-98000,-34300,-15876,-8624,-5200,-3375,6350400,-4762800,-882000,-308700,-142884,-77616,-46800,-30375,-20825
%N A120078 Coefficient triangle of numerator polynomials appearing in certain column o.g.f.s related to the H-atom spectrum.
%C A120078 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.
%C A120078 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.
%H A120078 G. C. Greubel, <a href="/A120078/b120078.txt">Rows n = 1..50 of the triangle, flattened</a>
%H A120078 Wolfdieter Lang, <a href="/A120078/a120078.txt">First ten rows</a>
%F A120078 T(n, k) = A051418(n) * (1 if k = 1 otherwise 1/k^2 - 1/(k-1)^2). - _G. C. Greubel_, Apr 26 2023
%e A120078 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)).
%e A120078 Triangle begins:
%e A120078        1;
%e A120078        4,      -3;
%e A120078       36,     -27,     -5;
%e A120078      144,    -108,    -20,    -7;
%e A120078     3600,   -2700,   -500,  -175,   -81;
%e A120078     3600,   -2700,   -500,  -175,   -81,   -44;
%e A120078   176400, -132300, -24500, -8575, -3969, -2156, -1300;
%t A120078 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 *)
%o A120078 (Magma)
%o A120078 f:= func< n | n eq 1 select 1 else 1/n^2 -1/(n-1)^2 >;
%o A120078 A120078:= func< n,k | (Lcm([1..n]))^2*f(k) >;
%o A120078 [A120078(n,k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Apr 26 2023
%o A120078 (SageMath)
%o A120078 def f(k): return 1 if (k==1) else 1/k^2 - 1/(k-1)^2
%o A120078 def A120078(n,k): return (lcm(range(1, n+1)))^2*f(k)
%o A120078 flatten([[A120078(n,k) for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Apr 26 2023
%Y A120078 Row sums (unsigned) give A120079.
%Y A120078 Signed row sums conjectured to coincide with A027451.
%Y A120078 Cf. A027451, A051418, A120072, A120073, A120079.
%K A120078 sign,tabl
%O A120078 1,2
%A A120078 _Wolfdieter Lang_, Jul 20 2006