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A318642
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G.f. C(x) satisfies: Sum_{n>=0} (-1)^n * n * ( C(C(x)) - (-1)^n*C(C(-x)) )^n = 0.
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4
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1, 2, 4, 56, 304, 2944, 22592, 196864, 1700352, 14416896, 127798272, 1141090304, 10314440704, 92351643648, 836353277952, 7885603078144, 74487690493952, 634585643614208, 5307259488829440, 71216920022482944, 950665921628733440, -1672932555803852800, -180293500222111219712, 4877207777660970729472, 164125283521717784805376, -3434577882064669615587328
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f. C(x) satisfies:
(1) C(-C(-x)) = x.
(2) 0 = Sum_{n>=0} (-1)^n * n * ( C(C(x)) - (-1)^n*C(C(-x)) )^n.
(3) 0 = Sum_{n>=0} n * ( x + (-1)^n*C(C(C(C(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 = C(C(C(C(x)))).
(5) C(x) = D(D(x)), the 2nd iteration of the g.f. D(x) of A318643.
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EXAMPLE
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G.f.: 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 + ...
where C(-C(-x)) = x.
RELATED SERIES.
(a) If C(C(C(C(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
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 + ... + A318641(n)*x^n + ...
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 +- ...
(c) If 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.
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PROG
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(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(HALF(x*Ser(A))), n)}
for(n=1, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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