A177395 G.f. satisfies: x = A(x) - A(A(x))^2 - A(A(A(x)))^3.

1, 1, 5, 37, 338, 3530, 40546, 500781, 6556080, 90097535, 1290778689, 19180015667, 294460699563, 4656776745569, 75682133890995, 1261603117268148, 21537605020132685, 376060923637721700, 6708681746445946648

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n), n= 1..100.

Formula

G.f. satisfies: x = A( x - A(x)^2 - A(A(x))^3 ).

...

G.f. satisfies: A_{n}(x) = A_{n+1}(x) - 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 + xB^2 + x^2*C^3;

. B = A + xC^2 + x^2*D^3;

. C = B + xD^2 + x^2*E^3;

. D = C + xE^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 + x^2 + 5*x^3 + 37*x^4 + 338*x^5 + 3530*x^6 +...
Coefficients in the iterations A_{n}(x), n=1..9, of A(x) begin:
A_1: [1, 1, 5, 37, 338, 3530, 40546, 500781, ...];
A_2: [1, 2, 12, 100, 998, 11197, 136682, 1780674, ...];
A_3: [1, 3, 21, 195, 2120, 25571, 332664, 4589974, ...];
A_4: [1, 4, 32, 328, 3868, 50078, 694340, 10157760, ...];
A_5: [1, 5, 45, 505, 6430, 89120, 1315126, 20388639, ...];
A_6: [1, 6, 60, 732, 10018, 148195, 2322702, 38106722, ...];
A_7: [1, 7, 77, 1015, 14868, 234017, 3886428, 67351872, ...];
A_8: [1, 8, 96, 1360, 21240, 354636, 6225480, 113733264, ...];
A_9: [1, 9, 117, 1773, 29418, 519558, 9617706, 184845297,...].
Coefficients in functions: x = A(x) - A_2(x)^2 - A_3(x)^3 begin:
(A_1)^1: [1, 1, 5, 37, 338, 3530, 40546, 500781, 6556080, ...];
(A_2)^2: [0, 1, 4, 28, 248, 2540, 28786, 352104, 4576404 ...];
(A_3)^3: [0, 0, 1,. 9,. 90,. 990, 11760, 148677, 1979676, ...].
Coefficients in functions: A(x) = A_2(x) - A_3(x)^2 - A_4(x)^3 begin:
(A_2)^1: [1, 2, 12, 100, 998, 11197, 136682, 1780674, 24453430, ...];
(A_3)^2: [0, 1,. 6,. 51, 516,. 5851,. 72052,. 945819, 13076714, ...];
(A_4)^3: [0, 0,. 1,. 12, 144,. 1816,. 24084,. 334074,. 4820636, ...].
Coefficients in functions: A_2(x) = A_3(x) - A_4(x)^2 -A_5(x)^3 begin:
(A_3)^1: [1, 3, 21, 195, 2120, 25571, 332664, 4589974, 66441348, ...];
(A_4)^2: [0, 1,. 8,. 80,. 912, 11384, 152092, 2144440, 31612640, ...];
(A_5)^3: [0, 0,. 1,. 15,. 210,. 2990,. 43890,. 664860, 10375278, ...].

Prog

(PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x-(A+x*O(x^n))^2-subst(A, x, A+x*O(x^n))^3)); polcoeff(A, n))}

Crossrefs

Cf. A139702, A177396, A171780.

Sequence in context: A199562 A246540 A365842 * A370343 A258296 A197713

Adjacent sequences: A177392 A177393 A177394 * A177396 A177397 A177398

Keyword

nonn

Author

Paul D. Hanna, May 29 2010

Extensions

Typos in examples corrected by Paul D. Hanna, May 29 2010

Formula corrected by Paul D. Hanna, May 29 2010

Status

approved