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

1, 1, 2, 8, 50, 382, 3434, 35694, 421682, 5582158, 81860978, 1317457646, 23087951666, 437673142510, 8924179990322, 194763818998638, 4530072136715954, 111870258525352174, 2923319958390174770, 80590596894930389102, 2337567736223817582002, 71162943130933082039278

Offset: 0

Author

Paul D. Hanna, Nov 07 2013

Keywords

• nonn

Comments

Compare to the identity: Sum_{n>=0} x^n*Product_{k=1..n} -(k + x)/(1 - k*x - x^2) = 1 - x.

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).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 50*x^4 + 382*x^5 + 3434*x^6 +...
where
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)) +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..275

Crossrefs

Cf. A231274.

Programs

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)}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) ~ n! / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Oct 30 2014

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