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A231352 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k - x) / (1 - k*x + x^2). 2
1, 1, 2, 8, 50, 382, 3434, 35694, 421682, 5582158, 81860978, 1317457646, 23087951666, 437673142510, 8924179990322, 194763818998638, 4530072136715954, 111870258525352174, 2923319958390174770, 80590596894930389102, 2337567736223817582002, 71162943130933082039278 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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).
LINKS
FORMULA
a(n) ~ n! / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Oct 30 2014
EXAMPLE
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)) +...
PROG
(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), ", "))
CROSSREFS
Cf. A231274.
Sequence in context: A027047 A034491 A360949 * A186182 A274273 A121677
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 07 2013
STATUS
approved

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Last modified March 4 23:31 EST 2024. Contains 370537 sequences. (Running on oeis4.)