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%I #14 Jul 15 2021 15:05:50
%S 1,3,46,1899,163476,25333590,6412369860,2473269931755,
%T 1379817056827720,1069150908119474628,1113779885682143602440,
%U 1518901247410616194635510,2651993653876241574715172280,5817640695573490720735010689620
%N -log( Sum_{n>=0} (-x)^n/n!^3 ) = Sum_{n>=1} a(n)*x^n/n!^3.
%F Equals column 0 of the matrix log of triangle T(n,k) = (-1)^(n-k)*C(n,k)^3.
%F a(n) = -(-1)^n + (1/n) * Sum_{k=1..n-1} (-1)^(n-k-1) * binomial(n,k)^3 * k * a(k). - _Ilya Gutkovskiy_, Jul 15 2021
%e L(x) = -log(1 - x + x^2/2!^3 - x^3/3!^3 + x^4/4!^3 - x^5/5!^3 +-...)
%e where
%e L(x) = x + 3*x^2/2!^3 + 46*x^3/3!^3 + 1899*x^4/4!^3 + 163476*x^5/5!^3 +...
%e ALTERNATE GENERATING METHOD.
%e A signed version of A181543(n,k) = C(n,k)^3 begins:
%e 1;
%e 1, 1;
%e 1, 8, 1;
%e 1, 27, 27, 1;
%e 1, 64, 216, 64, 1;
%e 1, 125, 1000, 1000, 125, 1; ...
%e The matrix log of triangle A181543 begins:
%e 0;
%e 1, 0;
%e -3, 8, 0;
%e 46, -81, 27, 0;
%e -1899, 2944, -648, 64, 0;
%e 163476, -237375, 46000, -3000, 125, 0; ...
%e in which this sequence (signed) is found in column 0.
%o (PARI) {a(n)=n!^3*polcoeff(-log(sum(m=0,n,(-x)^m/m!^3)+x*O(x^n)),n)}
%o (PARI) /* As Column 0 of the Matrix Log of signed Triangle A181543 */
%o {a(n)=local(L,M=matrix(n+1,n+1,r,c,if(r>=c,(-1)^(r-c)*binomial(r-1,c-1)^3)));
%o L=sum(n=1,#M,(M^0-M)^n/n);if(n<0,0,L[n+1,1])}
%Y Cf. A002190 (variant), A181543.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jul 26 2011
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