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A231352 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k - x) / (1 - k*x + x^2).

%I A231352 #11 Jul 14 2025 00:15:06
%S A231352 1,1,2,8,50,382,3434,35694,421682,5582158,81860978,1317457646,
%T A231352 23087951666,437673142510,8924179990322,194763818998638,
%U A231352 4530072136715954,111870258525352174,2923319958390174770,80590596894930389102,2337567736223817582002,71162943130933082039278
%N A231352 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k - x) / (1 - k*x + x^2).
%C A231352 Compare to the identity: Sum_{n>=0} x^n*Product_{k=1..n} -(k + x)/(1 - k*x - x^2) = 1 - x.
%C A231352 Compare also to the identity: Sum_{n>=0} x^n*Product_{k=1..n} (k + x)/(1 + k*x + x^2) = (1+x^2)/(1-x).
%H A231352 Vaclav Kotesovec, <a href="/A231352/b231352.txt">Table of n, a(n) for n = 0..275</a>
%F A231352 a(n) ~ n! / (2 * (log(2))^(n+1)). - _Vaclav Kotesovec_, Oct 30 2014
%F A231352 G.f. (conjecture): 1/2 + (1/2)*Sum_{n >= 0} (2*x)^n * Product_{k = 1..n} (k - x)/(1 + k*x). - _Peter Bala_, Jul 12 2025
%e A231352 G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 50*x^4 + 382*x^5 + 3434*x^6 +...
%e A231352 where
%e A231352 A(x) = 1 + x*(1-x)/(1-x+x^2) + x^2*(1-x)*(2-x)/((1-x+x^2)*(1-2*x+x^2)) + x^3*(1-x)*(2-x)*(3-x)/((1-x+x^2)*(1-2*x+x^2)*(1-3*x+x^2)) + x^4*(1-x)*(2-x)*(3-x)*(4-x)/((1-x+x^2)*(1-2*x+x^2)*(1-3*x+x^2)*(1-4*x+x^2)) +...
%o A231352 (PARI) {a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (k-x)/(1-k*x+x^2 +x*O(x^n))) ), n)}
%o A231352 for(n=0, 30, print1(a(n), ", "))
%Y A231352 Cf. A231274.
%K A231352 nonn
%O A231352 0,3
%A A231352 _Paul D. Hanna_, Nov 07 2013