A171780 G.f. F_1(x) satisfies: x = Sum_{n>=1} F_{n}(x)^n, where the n-th iteration of the g.f. is defined by: F_{n}(x) = F_{n-1}( F_1(x) ) with F_0(x) = x.

1, -1, 3, -12, 55, -273, 1431, -7837, 44726, -266381, 1658300, -10768609, 72407500, -498510748, 3477131466, -24461950817, 174793982029, -1294469021982, 10177610535232, -85391010070808, 741460526149745, -6291905077685633

Offset

1,3

Links

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

Example

G.f.: F_1(x) = x - x^2 + 3*x^3 - 12*x^4 + 55*x^5 - 273*x^6 + 1431*x^7 +...
Let F_{n}(x) denote the n-th iteration of g.f. F_1(x), then:
(1) x = F_1(x)^1 + F_2(x)^2 + F_3(x)^3 + F_4(x)^4 + F_5(x)^5 +...
(2) F_n(x) = F_{n+1}(x) + F_{n+2}(x)^2 + F_{n+3}(x)^3 + F_{n+4}(x)^4 +...
The initial iterations of the g.f. begin:
F_2(x) = x - 2*x^2 + 8*x^3 - 40*x^4 + 224*x^5 - 1345*x^6 +...
F_3(x) = x - 3*x^2 + 15*x^3 - 90*x^4 + 595*x^5 - 4184*x^6 +...
F_4(x) = x - 4*x^2 + 24*x^3 - 168*x^4 + 1280*x^5 - 10306*x^6 +...
F_5(x) = x - 5*x^2 + 35*x^3 - 280*x^4 + 2415*x^5 - 21895*x^6 +...
F_6(x) = x - 6*x^2 + 48*x^3 - 432*x^4 + 4160*x^5 - 41923*x^6 +...
F_7(x) = x - 7*x^2 + 63*x^3 - 630*x^4 + 6699*x^5 - 74270*x^6 +...
F_8(x) = x - 8*x^2 + 80*x^3 - 880*x^4 + 10240*x^5 - 123844*x^6 +...
where initial terms of F_n(x)^n for n=1..8 begin:
F_1(x)^1 = x - x^2 + 3*x^3 - 12*x^4 + 55*x^5 - 273*x^6 + 1431*x^7 +...
F_2(x)^2 = x^2 - 4*x^3 + 20*x^4 - 112*x^5 + 672*x^6 - 4226*x^7 +...
F_3(x)^3 = x^3 - 9*x^4 + 72*x^5 - 567*x^6 + 4485*x^7 - 35817*x^8 +...
F_4(x)^4 = x^4 - 16*x^5 + 192*x^6 - 2080*x^7 + 21504*x^8 +...
F_5(x)^5 = x^5 - 25*x^6 + 425*x^7 - 6150*x^8 + 81700*x^9 +...
F_6(x)^6 = x^6 - 36*x^7 + 828*x^8 - 15552*x^9 + 260400*x^10 +...
F_7(x)^7 = x^7 - 49*x^8 + 1470*x^9 - 34937*x^10 + 723632*x^11 +...
F_8(x)^8 = x^8 - 64*x^9 + 2432*x^10 - 71552*x^11 + 1802240*x^12 +...

Prog

(PARI) {a(n)=local(a_n=0, G=x, F=x-x^2+sum(k=3, n-1, a(k)*x^k)); if(n<1, 0, if(n==1, 1, for(k=2, n, G=x; for(i=1, k, G=subst(F, x, G+x*O(x^n))); a_n=a_n-polcoeff(G^k, n)); a_n))}

Crossrefs

Sequence in context: A007199 A179848 A001764 * A369159 A216493 A216494

Adjacent sequences: A171777 A171778 A171779 * A171781 A171782 A171783

Keyword

sign

Author

Paul D. Hanna, Jan 24 2010

Status

approved