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A212491
G.f.: A(x) = x + x*ITERATE^2(x + x*ITERATE^4(x + x*ITERATE^8(x + x*ITERATE^16(x + ...)))), where ITERATE^n(F(x)) denotes the n-th iteration of F(x), and the nesting of iterations continue indefinitely.
0
1, 1, 2, 10, 109, 2344, 95526, 7321508, 1062894966, 295821381776, 159750645147982, 169055785061863986, 353149667541498542122, 1463495730873823283553016, 12070551778811708797824867546, 198534598750957942596957876692320, 6519958358404237723197281898833187776
OFFSET
1,3
EXAMPLE
G.f.: A(x) = x + x^2 + 2*x^3 + 10*x^4 + 109*x^5 + 2344*x^6 + 95526*x^7 +...
where A(x) is generated by nesting 2^n-th iterations of shifted series:
A(x) = x + x*B(B(x));
B(x) = x + x*C(C(C(C(x))));
C(x) = x + x*D(D(D(D(D(D(D(D(x))))))));
D(x) = x + x*E(E(E(E(E(E(E(E(E(E(E(E(E(E(E(E(x)))))))))))); ...
The above series begin:
B(x) = x + x^2 + 4*x^3 + 44*x^4 + 1006*x^5 + 43392*x^6 + 3459214*x^7 +...
C(x) = x + x^2 + 8*x^3 + 184*x^4 + 8620*x^5 + 746176*x^6 + 117816292*x^7 +...
D(x) = x + x^2 + 16*x^3 + 752*x^4 + 71320*x^5 + 12374144*x^6 +...
E(x) = x + x^2 + 32*x^3 + 3040*x^4 + 580144*x^5 + 201551104*x^6 +...
PROG
(PARI) /* Define the (2^n)-th iteration of function F: */
{ITERATE2(n, F, p=#F)=local(G=F); for(i=1, n, G=subst(G, 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=ITERATE2(n-k, x + x*A, n)); polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 18 2012
STATUS
approved