OFFSET
0,3
FORMULA
Euler transform of the coefficients in A(x)/A(-x), where A(x) is the g.f. of this sequence.
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 5*x^3 + 14*x^4 + 28*x^5 + 77*x^6 + 173*x^7 +...
where
log(A(x)) = A(x)/A(-x)*x + A(x^2)/A(-x^2)*x^2/2 + A(x^3)/A(-x^3)*x^3/3 +...
more explicitly,
log(A(x)) = x + 5*x^2/2 + 7*x^3/3 + 29*x^4/4 + 51*x^5/5 + 191*x^6/6 + 407*x^7/7 + 1485*x^8/8 + 3409*x^9/9 + 12315*x^10/10 +...
This sequence equals the Euler transform of coefficients in A(x)/A(-x):
[1,2,2,6,10,30,58,182,378,1226,2658,8798,19634,65990,150338,511054,...];
A(x) = 1/((1-x) *(1-x^2)^2 *(1-x^3)^2 *(1-x^4)^6 *(1-x^5)^10 *(1-x^6)^30 *(1-x^7)^58 *(1-x^8)^182 *(1-x^9)^378 *...).
PROG
(PARI) {a(n)=local(A=1+x, B); for(i=1, n, B=(A/subst(A, x, -x)); A=exp(sum(m=1, n, subst(B, x, x^m+x*O(x^n))*x^m/m))); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 29 2011
STATUS
approved