%I #13 Sep 19 2024 05:53:21
%S 1,-1,4,-76,7124,-3326876,7760553124,-90490361296876,
%T 5275336666748203124,-1537656615631182860546876,
%U 2240970675863910673065189453124,-16329855533286908545970966339091796876,594974481262862479448134839533519744970703124
%N Column 0 of the matrix inverse of triangle A118190(n,k) = 5^(k*(n-k)).
%C The entire matrix inverse of triangle A118190 is determined by column 0 (this sequence): [A118190^-1](n,k) = a(n-k)*5^(k*(n-k)) for n>=k>=0. Any g.f. of the form: Sum_{k>=0} b(k)*x^k may be expressed as: Sum_{n>=0} c(n)*x^n/(1-5^n*x) by applying the inverse transformation: c(n) = Sum_{k=0..n} a(n-k)*b(k)*5^(k*(n-k)).
%H G. C. Greubel, <a href="/A118193/b118193.txt">Table of n, a(n) for n = 0..50</a>
%F G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1-5^n*x).
%F 0^n = Sum_{k=0..n} a(k)*5^(k*(n-k)) for n>=0.
%F a(n) = - Sum_{j=0..n-1} 5^(j*(n-j))*a(j) with a(0) = 1 and a(1) = -1. - _G. C. Greubel_, Jun 29 2021
%e Recurrence at n=4: 0 = a(0)*(5^0)^4 +a(1)*(5^1)^3 +a(2)*(5^2)^2 +a(3)*(5^3)^1 +a(4)*(5^4)^0 = 1*(5^0) - 1*(5^3) + 4*(5^4) - 76*(5^3) + 7124*(5^0).
%e The g.f. is illustrated by: 1 = 1/(1-x) - 1*x/(1-5*x) + 4*x^2/(1-25*x) - 76*x^3/(1-125*x) + 7124*x^4/(1-625*x) - 3326876*x^5/(1-3125*x) + 7760553124*x^6/(1-15625*x) +...
%t a[n_]:= a[n]= If[n<2, (-1)^n, -Sum[5^(j*(n-j))*a[j], {j,0, n-1}]];
%t Table[a[n], {n, 0, 30}] (* _G. C. Greubel_, Jun 29 2021 *)
%o (PARI) {a(n)=local(T=matrix(n+1,n+1,r,c,if(r>=c,(5^(c-1))^(r-c)))); return((T^-1)[n+1,1])}
%o (Sage)
%o @CachedFunction
%o def a(n): return (-1)^n if (n<2) else -sum(5^(j*(n-j))*a(j) for j in (0..n-1))
%o [a(n) for n in (0..30)] # _G. C. Greubel_, Jun 29 2021
%Y Cf. A118190.
%K sign
%O 0,3
%A _Paul D. Hanna_, Apr 15 2006