A372532 Expansion of g.f. A(x) satisfying A(x)^4 = A( x*A(x)^3/(1 - A(x)) ).

1, 1, 2, 5, 15, 48, 160, 549, 1930, 6919, 25200, 92976, 346757, 1305140, 4951216, 18912245, 72675115, 280761688, 1089800460, 4248151335, 16623220558, 65273436984, 257115848688, 1015721354400, 4023189912040, 15974444935191, 63571105091684, 253513322846012, 1012942417348605

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..520

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A(x)^4 = A( x*A(x)^3/(1 - A(x)) ).

(2) A(x)^16 = A( x*A(x)^15/((1 - A(x))*(1 - A(x)^4)) ).

(3) A(x)^64 = A( x*A(x)^63/((1 - A(x))*(1 - A(x)^4)*(1 - A(x)^16)) ).

(4) A(x)^(4^n) = A( x*A(x)^(4^n-1)/Product_{k=0..n-1} (1 - A(x)^(4^k)) ) for n > 0.

(5) A(x) = x / Product_{n>=0} (1 - A(x)^(4^n)).

(6) A(x) = Series_Reversion( x * Product_{n>=0} (1 - x^(4^n)) ).

(7) x = A(x) * Sum_{n>=0} (-1)^A010060(n) * A(x)^A000695(n), where A010060 is the Thue-Morse sequence and A000695 gives sums of distinct powers of 4.

The radius of convergence r and A(r) satisfy 1 = Sum_{n>=0} 4^n * A(r)^(4^n)/(1 - A(r)^(4^n)) and r = A(r) * Product_{n>=0} (1 - A(r)^(4^n)), where r = 0.23735646078435954136834955920887956765296150123281028... and A(r) = 0.45218226260527732381925578383609182019094441327410056...

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 48*x^6 + 160*x^7 + 549*x^8 + 1930*x^9 + 6919*x^10 + 25200*x^11 + 92976*x^12 + ...
where A(x)^4 = A( x*A(x)^3/(1 - A(x)) ).
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 28*x^6 + 93*x^7 + 321*x^8 + 1136*x^9 + 4092*x^10 + 14955*x^11 + 55328*x^12 + 206829*x^13 + ...
A(x)^4 = x^4 + 4*x^5 + 14*x^6 + 48*x^7 + 169*x^8 + 608*x^9 + 2222*x^10 + 8216*x^11 + 30680*x^12 + 115556*x^13 + 438554*x^14 + ...
x*A(x)^3/(1 - A(x)) = x^4 + 4*x^5 + 14*x^6 + 48*x^7 + 168*x^8 + 600*x^9 + 2178*x^10 + 8008*x^11 + 29762*x^12 + 111644*x^13 + ...
Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then
R(x) = x * Product_{n>=0} (1 - x^(4^n)) = x - x^2 - x^5 + x^6 - x^17 + x^18 + x^21 - x^22 - x^65 + x^66 + x^69 - x^70 + x^81 - x^82 - x^85 + x^86 - x^257 + x^258 + x^261 + ... + (-1)^A010060(n)*x^(A000695(n) + 1) + ...
thus,
x = A(x) * (1 - A(x)) * (1 - A(x)^4) * (1 - A(x)^16) * (1 - A(x)^64) * (1 - A(x)^256) * ... * (1 - A(x)^(4^n)) * ...
SPECIFIC VALUES.
A(t) = 2/5 at t = (2/5) * Product_{n>=0} (1 - (2/5)^(4^n)) = 0.2338558995596128026623999920422960979429704653...
A(t) = 1/3 at t = (1/3) * Product_{n>=0} (1 - 1/3^(4^n)) = 0.2194787328986396432551386254242908520274591882...
A(t) = 1/4 at t = (1/4) * Product_{n>=0} (1 - 1/4^(4^n)) = 0.1867675780815147845714818686246871948236698894...
A(t) = 1/5 at t = (1/5) * Product_{n>=0} (1 - 1/5^(4^n)) = 0.1597439999989531017215999999999999999999999997...
A(1/5) = 0.2791419705799491640241970731636463821918278265598702481...
A(1/6) = 0.2119087418184569371633725749849800368394048924883176302...
A(1/7) = 0.1728682698948146927220680877897385568988140527227279611...
A(1/6)^4 = A(t) at t = (1/6)*A(1/6)^3/(1 - A(1/6)) = 0.0020124210815...
A(1/7)^4 = A(t) at t = (1/7)*A(1/7)^3/(1 - A(1/7)) = 0.0008922224261..

Prog

(PARI) /* From Series_Reversion( x * Product_{n>=0} (1 - x^(4^n)) ) */
{a(n) = my(A, M=ceil(log(n+1)/log(4))); A = serreverse( x * prod(m=0, M, 1 - x^(4^m)) + x*O(x^n) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) /* From A(x)^4 = A( x*A(x)^3/(1 - A(x)) ) */
{a(n) = my(A=[0, 1], F); for(i=1, n, A = concat(A, 0); F=Ser(A);
A[#A] = polcoeff( subst(F, x, x*F^3/(1 - F) ) - F^4, #A+2) ); H=A; A[n+1]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A372530, A372531, A000695, A010060.

Sequence in context: A261003 A365268 A366041 * A218251 A203067 A278077

Adjacent sequences: A372529 A372530 A372531 * A372533 A372534 A372535

Keyword

nonn

Author

Paul D. Hanna, May 17 2024

Status

approved