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A318641 G.f. B(x) satisfies: Sum_{n>=0} (-1)^n * n * (B(x) - (-1)^n*B(-x))^n = 0. 4
1, 4, 16, 160, 1408, 13760, 140288, 1459200, 15595520, 168584192, 1847791616, 20524785664, 230327189504, 2605161103360, 29668221648896, 340391560216576, 3932276970749952, 45577849469665280, 529120083810713600, 6194461391984787456, 73492658811551350784, 862792881224493826048, 9756093957689676136448, 117188194561127830519808, 1695314645779792548331520 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

First negative term is a(31).

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..400

FORMULA

G.f. B(x) satisfies:

(1) B(-B(-x)) = x.

(2) 0 = Sum_{n>=0} (-1)^n * n * (B(x) - (-1)^n*B(-x))^n.

(3) 0 = Sum_{n>=0} n * (x + (-1)^n*B(B(x)))^n.

(4) 0 = (A-x)*(1 + (A-x)^2)/(1 - (A-x)^2)^2  -  2*(A+x)^2/(1 - (A+x)^2)^2, where A = B(B(x)).

(5) B(x) = D(D(D(D(x)))), the 4th iteration of the g.f. D(x) of A318643.

EXAMPLE

G.f.: B(x) = x + 4*x^2 + 16*x^3 + 160*x^4 + 1408*x^5 + 13760*x^6 + 140288*x^7 + 1459200*x^8 + 15595520*x^9 + 168584192*x^10 + 1847791616*x^11 + ...

such that

0 = (B(x) + B(-x)) - 2*(B(x) - B(-x))^2 + 3*(B(x) + B(-x))^3 - 4*(B(x) - B(-x))^4 + 5*(B(x) + B(-x))^5 - 6*(B(x) - B(-x))^6 + 7*(B(x) + B(-x))^7 - 8*(B(x) - B(-x))^8 + 9*(B(x) + B(-x))^9 - 10*(B(x) - B(-x))^10 +- ...

RELATED SERIES.

(a) If B(B(x)) = A(x) then

A(x) = x + 8*x^2 + 64*x^3 + 704*x^4 + 8704*x^5 + 113536*x^6 + 1544192*x^7 + 21671936*x^8 + 311468032*x^9 + 4560963584*x^10 + ... + A318640(n)*x^n + ...

such that

0 = (x - A(x)) + 2*(x + A(x))^2 + 3*(x - A(x))^3 + 4*(x + A(x))^4 + 5*(x - A(x))^5 + 6*(x + A(x))^6 + 7*(x - A(x))^7 + 8*(x + A(x))^8 + 9*(x - A(x))^9 + 10*(x + A(x))^10 + ...

(b) If C(C(x)) = B(x), then

C(x) = x + 2*x^2 + 4*x^3 + 56*x^4 + 304*x^5 + 2944*x^6 + 22592*x^7 + 196864*x^8 + 1700352*x^9 + 14416896*x^10 + 127798272*x^11 + 1141090304*x^12 + ... + A318642(n)*x^n + ...

where C(-C(-x)) = x.

(c) If D(D(D(D(x)))) = B(x), so that D(D(x)) = C(x), then

D(x) = x + x^2 + x^3 + 25*x^4 + 73*x^5 + 1025*x^6 + 4913*x^7 + 48985*x^8 + 311305*x^9 + 2393953*x^10 + 17903761*x^11 + 140986201*x^12 + 1096160649*x^13 + ... + A318643(n)*x^n + ...

where D(-D(-x)) = x.

PROG

(PARI) {HALF(F) = my(H=x); for(i=1, #F, H = (H + subst(F, x, serreverse(H +x*O(x^#F))))/2); H}

{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = polcoeff(sum(m=1, #A, m*(x + (-1)^m*x*Ser(A))^m), #A)); polcoeff( HALF(x*Ser(A)), n)}

for(n=1, 30, print1(a(n), ", "))

CROSSREFS

Cf. A318640, A318642, A318643.

Sequence in context: A005739 A279887 A226588 * A005741 A033911 A130691

Adjacent sequences:  A318638 A318639 A318640 * A318642 A318643 A318644

KEYWORD

sign

AUTHOR

Paul D. Hanna, Aug 31 2018

STATUS

approved

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Last modified December 9 15:04 EST 2019. Contains 329877 sequences. (Running on oeis4.)