A371712 Expansion of g.f. A(x) satisfying A( 2*x^2/(1-2*x)^2 ) = 2*A(x)^2.

1, 2, 6, 20, 74, 292, 1204, 5112, 22182, 97964, 439252, 1995864, 9175108, 42610024, 199643240, 942605936, 4480195078, 21417889996, 102907803268, 496639331384, 2406182074924, 11698276321976, 57051240809688, 279016001376400, 1368066809205340, 6723746064317432, 33118248167688264

Offset

1,2

Comments

Compare to D( x^2/(1-2*x^2)^2 ) = D(x)^2 where D(x) = C(x) - 1 and C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

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/(1+4*x) ) = -A(-x).

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

(3) A( 8*x^4/(1-4*x)^2 ) = 8*A(x)^4.

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

(5) A( 2*x^2/(1-x)^2 ) = 2*A( x/(1+x) )^2.

(6) A( 2*x^2/(1-3*x)^2 ) = 2*A( x/(1-x) )^2.

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

(8) A( 8*x^4/(1-4*x^2)^2 ) = 8*A( x/(1+2*x) )^4 = 2*A( 2*x^2 )^2.

(9) A(x/B(x)) = x where B(x) = 2*x + B(2*x^2)^(1/2) is the g.f. of A371711.

The radius of convergence is r = (3 - sqrt(5))/4 = 0.19098300562505257...

Example

G.f.: A(x) = x + 2*x^2 + 6*x^3 + 20*x^4 + 74*x^5 + 292*x^6 + 1204*x^7 + 5112*x^8 + 22182*x^9 + 97964*x^10 + 439252*x^11 + 1995864*x^12 + ...
where
A( 2*x^2/(1-2*x)^2 ) = 2*A(x)^2 and A( x/(1+4*x) ) = -A(-x).
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 16*x^4 + 64*x^5 + 264*x^6 + 1120*x^7 + 4864*x^8 + 21504*x^9 + 96416*x^10 + 437376*x^11 + 2004480*x^12 + ...
A( x/(1+2*x) ) = x + 2*x^3 + 10*x^5 + 60*x^7 + 422*x^9 + 3228*x^11 + 26052*x^13 + 217464*x^15 + 1861126*x^17 + ...
A( x/(1+2*x) )^2 = x^2 + 4*x^4 + 24*x^6 + 160*x^8 + 1184*x^10 + 9344*x^12 + 77056*x^14 + 654336*x^16 + ...
where A( x/(1+2*x) )^2 = A( 2*x^2 )/2.
If B(x) = x/Series_Reversion(A(x)), so that A(x/B(x)) = x, then
B(x) = 1 + 2*x + 2*x^2 + 2*x^4 - 4*x^6 + 22*x^8 - 36*x^10 - 108*x^12 + 376*x^14 + 1894*x^16 - 4180*x^18 - 10628*x^20 + ... + A371711(n)*x^n + ...
such that
1 = (...(((((((B(x) - 2*x)^2 - (2*x)^2)^2 - (2*x)^4)^2 - (2*x)^8)^2 - (2*x)^16)^2 - (2*x)^32)^2 - (2*x)^64)^2 - ...), an infinite nested square.
SPECIFIC VALUES.
A(1/6) = 0.29789229906054667355755647199664499026621078146355988...
A(1/7) = 0.22028240911749270058184659171007954537982971846657839...
A(1/8) = 2*A(1/6)^2 = 0.1774796436791563531387838244194094019267...
A(1/9) = 0.14936830726543771830506265617628097833926210696977323...
A(1/10) = 0.1292436104709804779603852368294850482664948994909500...
A(2/25) = 2*A(1/7)^2 = 0.097048679533212862523948544941373916211...
A(1/18) = 2*A(1/8)^2 = 0.062998047840960607307519834466070733930...
A(2/49) = 2*A(1/9)^2 = 0.044621782430684428529594838525892322559...
A(1/32) = 2*A(1/10)^2 = 0.03340782169514906928814772123377625594...

Prog

(PARI) {a(n) = my(A=[1], F); for(i=1, n, A=concat(A, 0); F=x*Ser(A);
A[#A] = polcoeff( subst(F, x, 2*x^2/(1-2*x)^2 ) - 2*F^2, #A+1)/4 ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A371711.

Sequence in context: A150155 A150156 A150157 * A145867 A188144 A245734

Adjacent sequences: A371709 A371710 A371711 * A371713 A371714 A371715

Keyword

nonn

Author

Paul D. Hanna, Apr 07 2024

Status

approved