A139702 G.f. satisfies: x = A( x + A(x)^2 ).

1, -1, 4, -24, 178, -1512, 14152, -142705, 1528212, -17211564, 202460400, -2474708496, 31310415376, -408815254832, 5495451727376, -75907303147652, 1075685334980240, -15618612118252960, 232102241507321384, -3526880759915999016

Offset

1,3

Comments

Signed version of A213591.

Links

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

Formula

Let G(x) = Series_Reversion( A(x) ) = x + A(x)^2, then G(x) = G(G(x)) - x^2 = g.f. of A138740.

G.f. satisfies: A(x) = x*G(-A(x)^2/x) where G(x) = 1 + x*G(1-1/G(x))^2 is the g.f. of A212411.

G.f.: A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:

A = 1 - x*B^2;

B = A - x*C^2;

C = B - x*D^2;

D = C - x*E^2;

E = D - x*F^2; ...

G.f. satisfies: A(x) = x*exp( Sum_{n>=0} (-1)^(n+1)*[d^n/dx^n A(x)^(2n+2)/x]/(n+1)! ). [Paul D. Hanna, Dec 18 2010]

Example

G.f.: A(x) = x - x^2 + 4*x^3 - 24*x^4 + 178*x^5 - 1512*x^6 +-...
A(x)^2 = x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-...
where A(x + A(x)^2) = x.
Let G(x) = Series_Reversion( A(x) ) = x + A(x)^2, then:
G(x) = x + x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 -+... and
G(G(x)) = x + 2*x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 -+...
so that G(x) = G(G(x)) - x^2 = g.f. of A138740.
Logarithmic series:
log(A(x)/x) = -A(x)^2/x + [d/dx A(x)^4/x]/2! - [d^2/dx^2 A(x)^6/x]/3! + [d^3/dx^3 A(x)^8/x]/4! -+...

Mathematica

nmax = 20; sol = {a[1] -> 1}; nmin = Length[sol]+1;
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[x - A[x + A[x]^2] + O[x]^(n+1), x][[nmin;; ]] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, nmin, nmax}];
a /@ Range[nmax] /. sol (* Jean-François Alcover, Nov 06 2019 *)

Prog

(PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x + (A+x*O(x^n))^2)); polcoeff(A, n))}
(PARI) /* n-th Derivative: */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
/* G.f.: [Paul D. Hanna, Dec 18 2010] */
{a(n)=local(A=x-x^2+x*O(x^n)); for(i=1, n,
A=x*exp(sum(m=0, n, (-1)^(m+1)*Dx(m, A^(2*m+2)/x)/(m+1)!)+x*O(x^n))); polcoeff(A, n)}

Crossrefs

Cf. A138740, A212411, A213591.

Cf. A088714, A088717, A091713, A120971, A140094, A140095.

Cf. A143426, A087949, A143435, A182969.

Sequence in context: A308543 A007846 A337507 * A213591 A243689 A309637

Adjacent sequences: A139699 A139700 A139701 * A139703 A139704 A139705

Keyword

sign

Author

Paul D. Hanna, Apr 30 2008, May 20 2008

Status

approved