OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = exp( Sum_{n>=1} (A(x)^n + A(-x)^n) * x^n/n ).
(2) A(x) = 1/((1 - x*A(x)) * (1 - x*A(-x))). - Paul D. Hanna, Dec 06 2009
(3) 0 = 1 - (3-x)*A(x) + (2-x)*A(x)^2 - (2-5*x)*x*A(x)^3 - (2+x)*x^2*A(x)^4 + 2*x^3*A(x)^5. - Paul D. Hanna, Feb 11 2024
EXAMPLE
G.f.: A(x) = 1 + 2*x + 3*x^2 + 10*x^3 + 27*x^4 + 112*x^5 + 336*x^6 + 1490*x^7 + 4791*x^8 + 22138*x^9 + 74079*x^10 + 351288*x^11 + 1207620*x^12 + ...
where the logarithm of A(x) may be written as
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 + (A(x)^4 + A(-x)^4)*x^4/4 + (A(x)^5 + A(-x)^5)*x^5/5 + ...
Incidentally, the square root of g.f. A(x) is an integer series starting
A(x)^(1/2) = 1 + x + x^2 + 4*x^3 + 9*x^4 + 43*x^5 + 108*x^6 + 558*x^7 + 1517*x^8 + 8175*x^9 + 23219*x^10 + 128516*x^11 + 375896*x^12 + ...
PROG
(PARI) {a(n) = my(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, (A^m+subst(A^m, x, -x)+x*O(x^n))*x^m/m))); polcoeff(A, n)}
(PARI) {a(n) = my(A=1+x); for(i=1, n, A=(1-x*A+x*O(x^n))^-1*(1-x*subst(A, x, -x)+x*O(x^n))^-1); polcoeff(A, n)} \\ Paul D. Hanna, Dec 06 2009
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 05 2009
STATUS
approved