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A208237
G.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (1 + k*x) / (1 + k*x + k^2*x^2).
6
1, 1, 2, 5, 15, 54, 223, 1045, 5474, 31685, 200895, 1384470, 10304431, 82376101, 703949762, 6403761365, 61784985615, 630180031734, 6775001385343, 76572619018165, 907658144193314, 11259399965148005, 145879271404693215, 1970471655222795990, 27702625497930064591
OFFSET
0,3
COMMENTS
Compare to the identity:
Sum_{n>=0} n! * x^n * Product_{k=1..n} (1 + x) / (1 + k*x + k*x^2) = 1/(1-x-x^2).
Compare also to the g.f. of A136127:
x*Sum_{n>=0} n! * x^n * Product_{k=1..n} (2 + k*x) / (1 + 2*k*x + k^2*x^2).
LINKS
FORMULA
a(n) ~ 2 * 3^(n/2 + 5/4) * n^(n+2) / (exp(n) * Pi^(n+3/2)). - Vaclav Kotesovec, Nov 02 2014
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 54*x^5 + 223*x^6 + 1045*x^7 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + 2!*x^2*(1+x)*(1+2*x)/((1+x+x^2)*(1+2*x+4*x^2)) + 3!*x^3*(1+x)*(1+2*x)*(1+3*x)/((1+x+x^2)*(1+2*x+4*x^2)*(1+3*x+9*x^2)) + 4!*x^4*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)/((1+x+x^2)*(1+2*x+4*x^2)*(1+3*x+9*x^2)*(1+4*x+16*x^2)) +...
PROG
(PARI) {a(n)=polcoeff( sum(m=0, n, m!*x^m*prod(k=1, m, (1+k*x)/(1+k*x+k^2*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 11 2013
STATUS
approved