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%I #9 Jun 30 2021 03:58:19
%S 0,1,0,-1,4,-11,-186,10823,-492536,5125897,10381552650,-6405856963573,
%T 3302055158456332,2338316177882689549,-30991279275364493410290,
%U 224870687441604081662836751,-1045565401111374322223949545456,-50507259999091315834370754855632879
%N Column 0 of the matrix log of triangle A117401, after term in row n is multiplied by n: a(n) = n*[log(A117401)](n,0), where A117401(n,k) = 2^(k*(n-k)).
%C The entire matrix log of triangle A117401 is determined by column 0 (this sequence): [log(A117401)](n,k) = a(n-k)*2^(k*(n-k))/(n-k) for n>k>=0.
%H G. C. Greubel, <a href="/A118197/b118197.txt">Table of n, a(n) for n = 0..80</a>
%F G.f.: x/(1-x)^2 = Sum_{n>=0} a(n)*x^n/(1-2^n*x).
%F By using the inverse transformation: a(n) = Sum_{k=0..n} k*A118196(n-k)*2^(k*(n-k)) for n>=0.
%F a(2^n) is divisible by 2^n.
%F G.f.: Sum_{n>=1} a(n)*x^n/[n*2^(n(n-1)/2)] = log(Sum_{n>=0} x^n/2^[n(n-1)/2]).
%e Column 0 of log(A117401) = [0, 1, 0, -1/3, 4/4, -11/5, -186/6, ...] and
%e consists of terms a(n)/n (n>0); these terms are integers at n = [0, 1, 2, 4, 6, 8, 10, 14, 16, 22, 26, 32, 34, 38, 46, 50, 58, 62, 64, 70, ...].
%e The g.f. is illustrated by:
%e x/(1-x)^2 = x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 +... = x/(1-2*x) - 0*x^2/(1-4*x) - 1*x^3/(1-8*x) + 4*x^4/(1-16*x) - 11*x^5/(1-32*x) - 186*x^6/(1-64*x) + 10823*x^7/(1-128*x) + ...
%e Define g.f.: G(x) = Sum_{n>=1} a(n)*x^n/[n * 2^(n(n-1)/2)], then G(x) = x + 0*x^2/4 - x^3/24 + 4*x^4/256 - 11*x^5/5120 - 186*x^6/196608 + ... and exp(G(x)) = 1 + x + x^2/2 + x^3/8 + x^4/64 + x^5/1024 + x^6/32768 + ...
%t A118196[n_]:= A118196[n]= If[n<2, (-1)^n, -Sum[2^(j*(n-j))*A118196[j], {j, 0, n-1}]];
%t a[n_]:= a[n]= -Sum[2^(j*(n-j))*j*A118196[j], {j, 0, n}];
%t Table[a[n], {n, 0, 30}] (* _G. C. Greubel_, Jun 30 2021 *)
%o (PARI) {a(n) = local(T=matrix(n+1,n+1,r,c,if(r>=c,(2^(c-1))^(r-c))), L=sum(m=1,#T,-(T^0-T)^m/m));return(n*L[n+1,1])};
%o (PARI) {a(n)=n*2^(n*(n-1)/2)*polcoeff(log(sum(k=0,n,x^k/2^(k*(k-1)/2))+x*O(x^n)),n)}
%o (Sage)
%o @CachedFunction
%o def A118196(n): return (-1)^n if (n<2) else -sum(2^(j*(n-j))*A118196(j) for j in (0..n-1))
%o def a(n): return (-1)*sum(2^(j*(n-j))*j*A118196(j) for j in (0..n))
%o [a(n) for n in (0..30)] # _G. C. Greubel_, Jun 30 2021
%Y Cf. A117401, A118196, A134531.
%K sign
%O 0,5
%A _Paul D. Hanna_, Apr 15 2006, Oct 30 2007
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