OFFSET
1,2
FORMULA
G.f. A(x) = Sum_{n>=1} a(n) * x^(6*n-5) satisfies:
(1) A( A(x) + A(x)^4 ) = G(x),
(2) G( A(x) - A(x)^4 ) = A(x),
(3) A( A(x) - A(x)^4 ) = -G(-x),
(4) A( A(x-x^4) + A(x-x^4)^4 ) = x,
where G(x) = x + G(x)^4 = Sum_{n>=1} binomial(4*n-3,n-1)/(4*n-3) * x^(3*n-2).
EXAMPLE
G.f.: A(x) = x + 2*x^7 + 35*x^13 + 812*x^19 + 21359*x^25 + 623244*x^31 + 18568947*x^37 + 638475040*x^43 + 13249877870*x^49 + 2024051330358*x^55 +...
such that Series_Reversion( A(x) + A(x)^4 ) = A(x) - A(x)^4, where
A(x)^4 = x^4 + 8*x^10 + 164*x^16 + 4120*x^22 + 113970*x^28 + 3416128*x^34 + 104776764*x^40 + 3565389600*x^46 + 88390775151*x^52 +...
A(x) + A(x)^4 = x + x^4 + 2*x^7 + 8*x^10 + 35*x^13 + 164*x^16 + 812*x^19 + 4120*x^22 + 21359*x^25 + 113970*x^28 + 623244*x^31 + 3416128*x^34 + 18568947*x^37 + 104776764*x^40 + 638475040*x^43 + 3565389600*x^46 + 13249877870*x^49 + 88390775151*x^52 + 2024051330358*x^55 +...
Also,
A( A(x) + A(x)^4 ) = x + x^4 + 4*x^7 + 22*x^10 + 140*x^13 + 969*x^16 + 7084*x^19 + 53820*x^22 +...+ binomial(4*n-3,n-1)/(4*n-3)*x^(3*n-2) +...
which is a g.f. of A002293.
Note that the following is an integer series:
sqrt(A(x)/x) = 1 + x^6 + 17*x^12 + 389*x^18 + 10146*x^24 + 294863*x^30 + 8741468*x^36 + 301536587*x^42 + 6008625027*x^48 + 994498807123*x^54 - 179176440388960*x^60 + 65367342797524884*x^66 - 27123073712119583646*x^72 +...
PROG
(PARI) {a(n) = my(Oxn=x*O(x^(6*n)), A = x +Oxn); for(i=1, 6*n, A = A + (x - subst(A + A^4, x, A - A^4 ))/2); polcoeff(A, 6*n-5)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Oct 12 2016
STATUS
approved