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A212922
G.f. satisfies: A(x) = x^2/(1-x) + Series_Reversion(x - x*A(x)).
3
1, 2, 5, 21, 120, 800, 5881, 46565, 391876, 3473879, 32226510, 311313683, 3119693862, 32333294383, 345754479372, 3807294710182, 43101806735623, 500977869387150, 5971566838065819, 72925079326977943, 911614856156206061, 11656341547670071145, 152347288068103795503
OFFSET
1,2
LINKS
FORMULA
G.f. A(x) also satisfies:
(1) A(x) = x/(1-x) + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*A(x)^n/n!.
(2) A(x) = x^2/(1-x) + x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*A(x)^n/n! ).
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 5*x^3 + 21*x^4 + 120*x^5 + 800*x^6 + 5881*x^7 +...
The series reversion of x - x*A(x) begins:
x + x^2 + 4*x^3 + 20*x^4 + 119*x^5 + 799*x^6 + 5880*x^7 +...
which equals A(x) - x^2/(1-x).
The g.f. A(x) satisfies:
A(x) - x^2/(1-x) = x + x*A(x) + d/dx x^2*A(x)^2/2! + d^2/dx^2 x^3*A(x)^3/3! + d^3/dx^3 x^4*A(x)^4/4! +...
log(A(x)/x - x/(1-x)) = A(x) + d/dx x*A(x)^2/2! + d^2/dx^2 x^2*A(x)^3/3! + d^3/dx^3 x^3*A(x)^4/4! +...
PROG
(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x^2/(1-x+x*O(x^n))+serreverse(x-x*A +x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
{a(n)=local(A=x+x^2); for(i=1, n, A=x/(1-x+x*O(x^n))+sum(m=1, n, Dx(m-1, x^m*A^m/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
{a(n)=local(A=x+x^2); for(i=1, n, A=x^2/(1-x+x*O(x^n))+x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*A^m/m!)+x*O(x^n)))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 31 2012
STATUS
approved