A118194 - OEIS

%I #9 Jun 30 2021 01:49:36

%S 0,1,-3,53,-4871,2262505,-5269940619,61424345593757,

%T -3580474937256484367,1043606492389898678125009,

%U -1520932783784930699920673828115,11082945991224258678496051788222656261,-403804307486446123171767495567989349951171863

%N Column 0 of the matrix log of triangle A118190, after term in row n is multiplied by n: a(n) = n*[log(A118190)](n,0), where A118190(n,k) = 5^(k*(n-k)).

%C The entire matrix log of triangle A118190 is determined by column 0 (this sequence): [log(A118190)](n,k) = a(n-k)5^(k*(n-k))/(n-k) for n>k>=0.

%H G. C. Greubel, <a href="/A118194/b118194.txt">Table of n, a(n) for n = 0..50</a>

%F G.f.: x/(1-x)^2 = Sum_{n>=0} a(n)*x^n/(1-5^n*x). By using the inverse transformation: a(n) = Sum_{k=0..n} k*A118193(n-k)*5^(k*(n-k)) for n>=0.

%e Column 0 of log(A118190) = [0, 1, -3/2, 53/3, -4871/4, ...].

%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 + 7*x^7 + ...

%e   = x/(1-5*x) -3*x^2/(1-25*x) +53*x^3/(1-125*x) -4871*x^4/(1-625*x) + 2262505*x^5/(1-3125*x) - 5269940619*x^6/(1-15625*x) + ...

%t A118193[n_]:= A118193[n]= If[n<2, (-1)^n, -Sum[5^(j*(n-j))*A118193[j], {j, 0, n-1}]];

%t a[n_]:= a[n]= -Sum[5^(j*(n-j))*j*A118193[j], {j, 0, n}];

%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))), L=sum(m=1,#T,-(T^0-T)^m/m));return(n*L[n+1,1])}

%o (Sage)

%o @CachedFunction

%o def A118193(n): return (-1)^n if (n<2) else -sum(5^(j*(n-j))*A118193(j) for j in (0..n-1))

%o def a(n): return (-1)*sum(5^(j*(n-j))*j*A118193(j) for j in (0..n))

%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