OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n), n= 1..100.
FORMULA
G.f. satisfies: x = A( x - 2*A(x)^2 + A(A(x))^3 ).
...
G.f. satisfies: A_{n}(x) = A_{n+1}(x) - 2*A_{n+2}(x)^2 + A_{n+3}(x)^3 where A_{n+1}(x) = A_{n}(A(x)) denotes iteration with A_0(x)=x.
...
Given g.f. A(x), A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
. A = 1 + 2xB^2 - x^2*C^3;
. B = A + 2xC^2 - x^2*D^3;
. C = B + 2xD^2 - x^2*E^3;
. D = C + 2xE^2 - x^2*F^3; ...
also B = A(A(x))/x, C = A(A(A(x)))/x, D = A(A(A(A(x))))/x, etc.
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 15*x^3 + 166*x^4 + 2253*x^5 + 34860*x^6 +...
Coefficients in the iterations A_{n}(x), n=1..9, of A(x) begin:
A_1: [1, 2, 15, 166, 2253, 34860, 592549, 10828938, ...];
A_2: [1, 4, 38, 490, 7473, 127274, 2349323, 46176042, ...];
A_3: [1, 6, 69, 1020, 17380, 325672, 6545871, 139035872, ...];
A_4: [1, 8, 108, 1804, 34078, 699716, 15287390, 350846310, ...];
A_5: [1, 10, 155, 2890, 60055, 1344140, 31807669, 786868272, ...];
A_6: [1, 12, 210, 4326, 98183, 2382590, 60814113, 1616326636, ...];
A_7: [1, 14, 273, 6160, 151718, 3971464, 108878847, 3097957506, ...];
A_8: [1, 16, 344, 8440, 224300, 6303752, 184875900, 5611606932, ...];
A_9: [1, 18, 423, 11214, 319953, 9612876, 300464469, 9696526206,...].
Coefficients in functions: x = A(x) - 2*A_2(x)^2 + A_3(x)^3 begin:
(A_1)^1: [1, 2, 15, 166, 2253, 34860, 592549, 10828938, ...];
(A_2)^2: [0, 1,. 8,. 92, 1284, 20310, 351572,. 6524886, ...];
(A_3)^3: [0, 0,. 1,. 18,. 315,. 5760, 110595,. 2220834, ...].
Coefficients in functions: A(x) = A_2(x) - 2*A_3(x)^2 +A_4(x)^3 begin:
(A_2)^1: [1, 4, 38, 490, 7473, 127274, 2349323, 46176042, ...];
(A_3)^2: [0, 1, 12, 174, 2868,. 51761, 1000664, 20438646, ...];
(A_4)^3: [0, 0,. 1,. 24,. 516,. 11108,. 244554,. 5530188, ...].
Coefficients in functions: A_2(x) = A_3(x) -2*A_4(x)^2 +A_5(x)^3 begin:
(A_3)^1: [1, 6, 69, 1020, 17380, 325672, 6545871, 139035872, ...];
(A_4)^2: [0, 1, 16,. 280,. 5336, 108684, 2334344,. 52385500, ...];
(A_5)^3: [0, 0,. 1,.. 30,.. 765,. 18970,. 472140,. 11911170, ...].
PROG
(PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - 2*(A+x*O(x^n))^2+subst(A, x, A+x*O(x^n))^3)); polcoeff(A, n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 29 2010
EXTENSIONS
Formula corrected by Paul D. Hanna, May 29 2010
STATUS
approved