A376230 Expansion of g.f. A(x) satisfying A(x)^2 = A( x*A(x) + x*A(x)^2 + x*A(x)^3 ).

1, 1, 3, 8, 28, 98, 370, 1423, 5643, 22753, 93299, 387324, 1625768, 6886156, 29399430, 126377000, 546527682, 2376094442, 10379414436, 45532904886, 200511864604, 886055084460, 3927826810396, 17462128520246, 77838085223570, 347813389031746, 1557683052973482, 6990670698115144

Offset: 1

Author

Paul D. Hanna, Sep 23 2024

Keywords

• nonn

Comments

Compare to C(x)^2 = C( x*C(x) + x*C(x)^2 ) where C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108).

Compare to D(x)^2 = D( x*D(x) + 2*x*D(x)^2 + x*D(x)^3 ) where D(x) = x*F(x)^2 and F(x) = 1 + F(x)^3 is the g.f. of A001764.

It appears that, for n >= 1, a(n) is odd iff n = 4*A000695(k) + {0,1,2,3} for some k >= 0, where A000695 is the Moser-de Bruijn sequence (sums of distinct powers of 4).

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 8*x^4 + 28*x^5 + 98*x^6 + 370*x^7 + 1423*x^8 + 5643*x^9 + 22753*x^10 + 93299*x^11 + 387324*x^12 + ...
where A(x)^2 = A( x*A(x) + x*A(x)^2 + x*A(x)^3 ).
RELATED SERIES.
Let B(x) be the g.f. of Stern's diatomic series (A002487):
B(x) = x + x^2 + 2*x^3 + x^4 + 3*x^5 + 2*x^6 + 3*x^7 + x^8 + 4*x^9 + 3*x^10 + 5*x^11 + 2*x^12 + 5*x^13 + 3*x^14 + 4*x^15 + x^16 + ...
then A(x)^2 = x * B( A(x) ) and A( x^2/B(x) ) = x.
Other related series begin as follows.
A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 22*x^5 + 81*x^6 + 300*x^7 + 1168*x^8 + 4622*x^9 + 18704*x^10 + 76738*x^11 + 319054*x^12 + ...
A(x)^3 = x^3 + 3*x^4 + 12*x^5 + 43*x^6 + 168*x^7 + 657*x^8 + 2649*x^9 + 10803*x^10 + 44733*x^11 + 187130*x^12 + ...
SPECIFIC VALUES.
A(t) = 2/5 at t = 0.21059927171685797509442589615351221149308548505349232763302...
A(t) = 1/3 at t = 0.20284350479378267499481171039846064829566135845385597426623...
A(t) = 1/4 at t = 0.17791461653470954766421766118399907657065676113208935616640...
A(t) = 1/5 at t = 0.15460046705113920070261079885331294477343970681952336915318...
A(1/5) = 0.31995483821441278259163540295892975411660207078522958569307...
A(1/6) = 0.22418805161328879302723377308422267982113532037722470920139...
A(1/7) = 0.17887744157158359437462802053243127220002079340556427992475...
A(1/8) = 0.15001315387877904231502214457445835910409883718703588438530...
A(1/10) = 0.11423875178408085947774841103426888472717517658954942399943...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..1030

Crossrefs

Cf. A002487, A371709, A371716.

Programs

PARI

/* Series_Reversion(x / Product_{n>=0} (1 + x^(2^n) + x^(2^(n+1))) ) */
{a(n) = my(A = serreverse(x / prod(k=0, #binary(n), (1 + x^(2^k) + x^(2^(k+1))) +x*O(x^n))));  polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

PARI

/* A(x)^2 = A( x*A(x) + x*A(x)^2 + x*A(x)^3 ) */
{a(n) = my(A=[1], F); for(i=1, n, A=concat(A, 0); F=x*Ser(A);
A[#A] = -polcoeff( F^2 - subst(F, x, x*F + x*F^2 + x*F^3), #A+1) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

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

(1) A(x)^2 = A( x*A(x) + x*A(x)^2 + x*A(x)^3 ).

(2) A(x)^4 = A( x*A(x)^3 * (1 + A(x) + A(x)^2) * (1 + A(x)^2 + A(x)^4) ).

(3) A(x)^8 = A( x*A(x)^7 * (1 + A(x) + A(x)^2) * (1 + A(x)^2 + A(x)^4) * (1 + A(x)^4 + A(x)^8) ).

(4) A(x)^(2^n) = A( x*A(x)^(2^n-1) * Product_{k=0..n-1} (1 + A(x)^(2^k) + A(x)^(2^(k+1))) ) for n > 0.

(5) A(x) = x * Product_{n>=0} (1 + A(x)^(2^n) + A(x)^(2^(n+1))).

(6) A(x) = Series_Reversion(x / Product_{n>=0} (1 + x^(2^n) + x^(2^(n+1))) ).

(7) A( x^2/B(x) ) = x where B(x) is the g.f. of Stern's diatomic series (A002487).

(8) A(x)^2 = x * B( A(x) ) where B(x) is the g.f. of Stern's diatomic series (A002487).

(9) A(x)^2 = x * Sum_{n>=0} A(x)^n * Sum_{k=0..n-1} (binomial(k, n-k-1) mod 2).

The radius of convergence r of g.f. A(x) and A(r) satisfy 1 = Sum_{n>=0} (2^n*A(r)^(2^n) + 2^(n+1)*A(r)^(2^(n+1))) / (1 + A(r)^(2^n) + A(r)^(2^(n+1))) and r = A(r) / Product_{n>=0} (1 + A(r)^(2^n) + A(r)^(2^(n+1))), where r = 0.210913447825795710516245118118286786032842961511008744076383999... and A(r) = 0.416616392852528289560364740676108057746635416495044590336240813...