OFFSET
1,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..100
EXAMPLE
G.f.: A(x) = x + x^2 + 2*x^3 + 8*x^4 + 52*x^5 + 480*x^6 + 5811*x^7 +...
where A(x) is generated by nested iterations of shifted series:
A(x) = x + x*B(B(x)), where
B(x) = x + x^2 + 3*x^3 + 18*x^4 + 165*x^5 + 2034*x^6 + 31500*x^7 +...;
B(x) = x + x*C(C(C(x))), where
C(x) = x + x^2 + 4*x^3 + 32*x^4 + 380*x^5 + 5904*x^6 + 112528*x^7 +...;
C(x) = x + x*D(D(D(D(x)))), where
D(x) = x + x^2 + 5*x^3 + 50*x^4 + 730*x^5 + 13720*x^6 + 311730*x^7 +...;
D(x) = x + x*E(E(E(E(E(x))))), where
E(x) = x + x^2 + 6*x^3 + 72*x^4 + 1248*x^5 + 27552*x^6 + 728175*x^7 +...;
E(x) = x + x*F(F(F(F(F(F(x)))))), where
F(x) = x + x^2 + 7*x^3 + 98*x^4 + 1967*x^5 + 49910*x^6 + 1505546*x^7 +...;
F(x) = x + x*G(G(G(G(G(G(G(x))))))), where
G(x) = x + x^2 + 8*x^3 + 128*x^4 + 2920*x^5 + 83744*x^6 + 2840520*x^7 +...;
G(x) = x + x*H(H(H(H(H(H(H(H(x)))))))), where
H(x) = x + x^2 + 9*x^3 + 162*x^4 + 4140*x^5 + 132444*x^6 + 4991148*x^7 +...; ...
PROG
(PARI) /* Define the n-th iteration of function F: */
{ITERATE(n, F, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}
/* A(x) results from nested iterations of shifted series: */
{a(n)=local(A=x); for(k=0, n, A=ITERATE(n-k+1, x + x*A, n)); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 11 2011
STATUS
approved