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A118188 Column 0 of the matrix inverse of triangle A118185(n,k) = (4^k)^(n-k).

%I A118188 #11 Jun 09 2025 04:53:42
%S A118188 1,-1,3,-33,1407,-237057,158992383,-425715556353,4556004503093247,
%T A118188 -194971932801554579457,33370662957719457037287423,
%U A118188 -22845215336421444625717664940033,62557106610069521429900219032249827327,-685195337175488637158242110253091749621661697
%N A118188 Column 0 of the matrix inverse of triangle A118185(n,k) = (4^k)^(n-k).
%C A118188 The entire matrix inverse of triangle A118185 is determined by column 0 (this sequence): [A118185^-1](n,k) = a(n-k)*4^(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/(1-4^n*x) by applying the inverse transformation: c(n) = Sum_{k=0..n} a(n-k)*b(k)*4^(k*(n-k)).
%H A118188 G. C. Greubel, <a href="/A118188/b118188.txt">Table of n, a(n) for n = 0..55</a>
%F A118188 G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1-4^n*x).
%F A118188 0^n = Sum_{k=0..n} a(k)*4^(k*(n-k)) for n>=0.
%F A118188 G.f.: Sum_{n>=0} a(n)*x^n/2^(n^2) = 1/Sum_{n>=0} x^n/2^(n^2). - _Vladeta Jovovic_, Oct 14 2009
%F A118188 a(n) = (-1)*Sum_{j=0..n-1} 4^(j*(n-j))*a(j) with a(0) = 1, and a(1) = -1. - _G. C. Greubel_, Jun 29 2021
%F A118188 a(n) ~ (-1)^n * c * d^n * 2^(n^2), where d = 0.3264264803687260917234459979520857231773098381012486773185370058835770869... and c = 1.9166236268503386759188959219480676595889651334615223163023463792603010986... - _Vaclav Kotesovec_, Jun 09 2025
%e A118188 Recurrence at n=4:
%e A118188 0 = a(0)*(4^0)^4 +a(1)*(4^1)^3 +a(2)*(4^2)^2 +a(3)*(4^3)^1 +a(4)*(4^4)^0
%e A118188 = 1*(4^0) - 1*(4^3) + 3*(4^4) - 33*(4^3) + 1407*(4^0).
%e A118188 The g.f. is illustrated by:
%e A118188 1 = 1/(1-x) - 1*x/(1-4*x) + 3*x^2/(1-16*x) - 33*x^3/(1-64*x) +
%e A118188 1407*x^4/(1-256*x) - 237057*x^5/(1-1024*x) + 158992383*x^6/(1-4096*x) +...
%t A118188 a[n_]:= a[n]= If[n<2, (-1)^n, -Sum[4^(j*(n-j))*a[j], {j, 0, n-1}]];
%t A118188 Table[a[n], {n, 0, 30}] (* _G. C. Greubel_, Jun 29 2021 *)
%o A118188 (PARI) {a(n)=local(T=matrix(n+1,n+1,r,c,if(r>=c,(4^(c-1))^(r-c)))); return((T^-1)[n+1,1])}
%o A118188 (Sage)
%o A118188 @CachedFunction
%o A118188 def a(n): return (-1)^n if (n<2) else -sum(4^(j*(n-j))*a(j) for j in (0..n-1))
%o A118188 [a(n) for n in (0..30)] # _G. C. Greubel_, Jun 29 2021
%Y A118188 Cf. A118185 (triangle).
%K A118188 sign
%O A118188 0,3
%A A118188 _Paul D. Hanna_, Apr 15 2006