A354124 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * (1 + x^n*A(x))^n / (1 + x^(n+1)*A(x))^(n+1).

1, 3, 7, 17, 37, 77, 143, 218, 205, -260, -2120, -7359, -18850, -36275, -38243, 75763, 636582, 2518615, 7562260, 18326392, 33734309, 29439139, -106574279, -745021520, -2876904500, -8612026926, -20904359944, -38363726381, -31422406713, 137039078742, 925573761117

Offset

0,2

Comments

The signs of the coefficients seem to fall into a cycle of period 14:

sign( a(7*n+k) ) = (-1)^n for k=1..7, n > 1 (conjecture).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..310

Example

G.f.: A(x) = 1 + 3*x + 7*x^2 + 17*x^3 + 37*x^4 + 77*x^5 + 143*x^6 + 218*x^7 + 205*x^8 - 260*x^9 - 2120*x^10 - 7359*x^11 - 18850*x^12 - 36275*x^13 - 38243*x^14 + 75763*x^15 + ...
such that
1 = 1/(1 + x*A(x)) + x*(1 + x*A(x))/(1 + x^2*A(x))^2 + x^2*(1 + x^2*A(x))^2/(1 + x^3*A(x))^3 + x^3*(1 + x^3*A(x))^3/(1 + x^4*A(x))^4 + x^4*(1 + x^4*A(x))^4/(1 + x^5*A(x))^5 + x^5*(1 + x^5*A(x))^5/(1 + x^6*A(x))^6 + ...
which, upon setting A = A(x), is equivalent to
1 = 1 + (1 - A)*x + (1 + A + A^2)*x^2 + (1 - 2*A - A^3)*x^3 + (1 + 2*A - 2*A^2 + A^4)*x^4 + (1 - 3*A + 3*A^2 - A^5)*x^5 + (1 + 3*A + A^2 + 3*A^3 + A^6)*x^6 + (1 - 4*A - 6*A^2 - 4*A^3 - A^7)*x^7 + (1 + 4*A + 6*A^2 - 4*A^4 + A^8)*x^8 + (1 - 5*A + 3*A^2 - 3*A^3 + 5*A^4 - A^9)*x^9 + (1 + 5*A - 12*A^2 + 12*A^3 + 5*A^5 + A^10)*x^10 + (1 - 6*A + 10*A^2 - 10*A^3 - 6*A^5 - A^11)*x^11 + (1 + 6*A + 6*A^2 + A^3 + 6*A^4 - 6*A^6 + A^12)*x^12 + (1 - 7*A - 20*A^2 - 12*A^3 - 20*A^4 + 7*A^6 - A^13)*x^13 + (1 + 7*A + 15*A^2 + 30*A^3 + 15*A^4 + 7*A^7 + A^14)*x^14 + (1 - 8*A + 10*A^2 - 20*A^3 - 10*A^5 - 8*A^7 - A^15)*x^15 + (1 + 8*A - 30*A^2 + 4*A^3 - 4*A^4 + 30*A^5 - 8*A^8 + A^16)*x^16 + ...
PATTERN OF SIGNS.
The signs (+-1) of the terms begin:
[+, +, +, +, +, +, +, +,  +, -, -, -, -, -, -,  +, +, +, +, +, +, +,  -, -, -, -, -, -, -,  +, +, +, +, +, +, +,  -, -, -, -, -, -, -,  +, +, +, +, +, +, +,  -, -, -, -, -, -, -,  +, +, +, +, +, +, +,  -, -, -, -, -, -, -, +, +, +, +, +, +, +,  -, -, -, -, -, -, -,  +, +, +, +, +, +, +,  -, -, -, -, -, -, -,  +, +, +, +, +, +, +,  -, ...]
which seems to follow the rule: sign( a(7*n+k) ) = (-1)^n for k=1..7, n > 1.

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, x^m * (1 + x^m*Ser(A))^m / (1 + x^(m+1)*Ser(A))^(m+1) ), #A)); H=A; A[n+1]}
for(n=0, 60, print1(a(n), ", "))

Crossrefs

Sequence in context: A295146 A014314 A221792 * A089099 A089116 A244629

Adjacent sequences: A354121 A354122 A354123 * A354125 A354126 A354127

Keyword

sign

Author

Paul D. Hanna, May 17 2022

Status

approved