A193420 -log( Sum_{n>=0} (-x)^n/n!^3 ) = Sum_{n>=1} a(n)*x^n/n!^3.

1, 3, 46, 1899, 163476, 25333590, 6412369860, 2473269931755, 1379817056827720, 1069150908119474628, 1113779885682143602440, 1518901247410616194635510, 2651993653876241574715172280, 5817640695573490720735010689620

Offset: 1

Author

Paul D. Hanna, Jul 26 2011

Keywords

• nonn

Examples

			L(x) = -log(1 - x + x^2/2!^3 - x^3/3!^3 + x^4/4!^3 - x^5/5!^3 +-...)
where
L(x) = x + 3*x^2/2!^3 + 46*x^3/3!^3 + 1899*x^4/4!^3 + 163476*x^5/5!^3 +...
ALTERNATE GENERATING METHOD.
A signed version of A181543(n,k) = C(n,k)^3 begins:
1;
1, 1;
1, 8, 1;
1, 27, 27, 1;
1, 64, 216, 64, 1;
1, 125, 1000, 1000, 125, 1; ...
The matrix log of triangle A181543 begins:
0;
1, 0;
-3, 8, 0;
46, -81, 27, 0;
-1899, 2944, -648, 64, 0;
163476, -237375, 46000, -3000, 125, 0; ...
in which this sequence (signed) is found in column 0.
		

Crossrefs

Cf. A002190 (variant), A181543.

Programs

PARI

{a(n)=n!^3*polcoeff(-log(sum(m=0,n,(-x)^m/m!^3)+x*O(x^n)),n)}

PARI

/* As Column 0 of the Matrix Log of signed Triangle A181543 */
{a(n)=local(L,M=matrix(n+1,n+1,r,c,if(r>=c,(-1)^(r-c)*binomial(r-1,c-1)^3)));
L=sum(n=1,#M,(M^0-M)^n/n);if(n<0,0,L[n+1,1])}

Formula

Equals column 0 of the matrix log of triangle T(n,k) = (-1)^(n-k)*C(n,k)^3.

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