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%I #3 Mar 30 2012 18:37:03
%S 1,1,0,-3,-10,-35,-186,-1162,-6980,-37893,-170170,-420926,2820168,
%T 58820034,648789218,5870211150,49367781216,424549221251,4031944331166,
%U 42858283306334,485093040406600,5516989209285204,60784199053120378,635030292370785486,6149124209685347592
%N E.g.f.: A(x) = Product_{n>=0} { exp(x)*[Sum_{k=0..n} (-x)^k/k! ] }.
%C Sum of series at x=1 converges to zero: 0 = 1 +1 +0/2! -3/3! -10/4! -35/5! +... G.f. for A129482 is closely related.
%F Special values: A(1) = 0.
%e E.g.f.: A(x) = 1 +x +0x^2/2! -3x^3/3! -10x^4/4! -35x^5/5! -186x^6/6!
%e -1162x^7/7! -6980x^8/8! -37893x^9/9! -170170x^10/10! -420926x^11/11! +...
%e Product formula is illustrated by:
%e A(x) = [exp(x)*(1)]*[exp(x)*(1 - x)]*[exp(x)*(1 - x + x^2/2!)]*
%e [exp(x)*(1 - x + x^2/2! - x^3/3!)]*
%e [exp(x)*(1 - x + x^2/2! - x^3/3! + x^4/4!)]*
%e [exp(x)*(1 - x + x^2/2! - x^3/3! + x^4/4! - x^5/5!)]*...*
%e [exp(x)*(Sum_{k=0..n} (-x)^k/k!) ]*...
%e Equivalently:
%e A(x) = [1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! +...]*
%e [1 - x^2/2! - 2x^3/3! - 3x^4/4! - 4x^5/5! - 5x^6/6! -...]*
%e [1 + x^3/3! + 3x^4/4! + 6x^5/5! + 10x^6/6! + 15x^7/7! +...]*
%e [1 - x^4/4! - 4x^5/5! - 10x^6/6! - 20x^7/7! - 35x^8/8! -...]*
%e [1 + x^5/5! + 5x^6/6! + 15x^7/7! + 35x^8/8! + 70x^9/9! +...]*...*
%e [1 + (-1)^n*Sum_{k>=0} C(n+k-1,n)*x^(n+k)/(n+k)! ]*...
%o (PARI) {a(n)=n!*polcoeff(prod(k=0,n,exp(x+x*O(x^n))*sum(j=0,k,(-x)^j/j!)),n)} /* Alternate Formula: */ {a(n)=n!*polcoeff(prod(k=0,n,1+(-1)^k*sum(i=1,n-k+1,binomial(k+i-1,k)*x^(k+i)/(k+i)! +x*O(x^n))),n)}
%Y Cf. A129482.
%K sign
%O 0,4
%A _Paul D. Hanna_, Apr 17 2007
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