A356781 Expansion of g.f. A(x) satisfying A(x) = A( x^2 + 2*x^2*A(x) )^(1/2), with A(0)=0, A'(0)=1.

1, 1, 1, 2, 4, 7, 14, 32, 74, 172, 408, 978, 2349, 5662, 13737, 33568, 82596, 204618, 510208, 1279544, 3224828, 8162144, 20735397, 52848816, 135088609, 346214873, 889451320, 2290164276, 5908894762, 15274778235, 39555942836, 102603159040, 266545251022

Offset: 1

Author

Paul D. Hanna, Aug 27 2022

Keywords

• nonn

Comments

Compare the g.f. to the following identities:

(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),

(2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),

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

Examples

			G.f. A(x) = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 14*x^7 + 32*x^8 + 74*x^9 + 172*x^10 + 408*x^11 + 978*x^12 + 2349*x^13 + 5662*x^14 + ...
where A(x)^2 = A( x^2 + 2*x^2*A(x) ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 26*x^7 + 54*x^8 + 122*x^9 + 284*x^10 + 668*x^11 + 1597*x^12 + 3864*x^13 + 9394*x^14 + ...
x^2 + 2*x^2*A(x) = x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 8*x^7 + 14*x^8 + 28*x^9 + 64*x^10 + 148*x^11 + 344*x^12 + 816*x^13 + 1956*x^14 + ...
Let G(x) = Series_Reversion( A(x) ) then
G(x) = x - x^2 + x^3 - 2*x^4 + 4*x^5 - 7*x^6 + 12*x^7 - 23*x^8 + 45*x^9 - 84*x^10 + 157*x^11 - 302*x^12 + 584*x^13 - 1121*x^14 + ...
where G(x)^2 = G(x^2)/(1 + 2*x) and G(A(x)) = x.
Also, the series bisections of G(x) = B1(x) - B2(x) begin
B1(x) = x + x^3 + 4*x^5 + 12*x^7 + 45*x^9 + 157*x^11 + 584*x^13 + 2155*x^15 + 8110*x^17 + ... + A370540(n)*x^(2*n+1) + ...
B2(x) = x^2 + 2*x^4 + 7*x^6 + 23*x^8 + 84*x^10 + 302*x^12 + 1121*x^14 + 4175*x^16 + 15739*x^18 + ...
where B2(x)/B1(x) = (1 - sqrt(1 - 4*x^2))/(2*x) = x*C(x^2) where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).
SPECIFIC VALUES.
A(sqrt(8)/8) = 0.655524504794958661...
A(1/3) = 0.562844396405786198664639714324421729...
A(1/4) = 0.343586921250887627089213663537057899...
A(1/5) = 0.253260720427093363870348121973670974...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..960

Crossrefs

Cf. A370540, A000108.

Programs

PARI

{a(n) = my(A=[1],G); for(i=1,n, A=concat(A,0); G = x*Ser(A); A = Vec((subst(G,x, x^2 + 2*x^2*G) +x^2*O(x^#A))^(1/2)); ); A[n]}
for(n=1, 40, print1(a(n), ", "))

Formula

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

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

(2) G(x) = sqrt( G(x^2)/(1 + 2*x) ), where A(G(x)) = x.

(3) G(x) = x*F(x^2)*(1 - x*C(x^2)) = x*F(x)^2*(1 - 4*x)/(1 - x*C(x)) where G(A(x)) = x, F(x) is the g.f. of A370540, and C(x) = (1 - sqrt(1-4*x))/(2*x) is the Catalan function (A000108). - Paul D. Hanna, Mar 12 2024

a(n) ~ c * d^n / n^(3/2), where d = 2.721808159464577... and c = 0.218334153814... - Vaclav Kotesovec, Mar 14 2024