A264231 G.f. satisfies: A(x) = A( -x/(1-4*x) ) / sqrt(1-4*x), with A(0)=1.

1, 1, 3, 10, 37, 144, 582, 2418, 10266, 44353, 194395, 862308, 3864100, 17466055, 79537815, 364543920, 1680205044, 7782321357, 36202366293, 169056409476, 792157776138, 3723252477558, 17548166926908, 82913613567012, 392650956009063, 1863327440038659, 8859299215070493, 42196115735716894, 201303626788647712, 961803888607574670

Offset: 0

Author

Paul D. Hanna, Nov 29 2015

Keywords

• nonn

Comments

Self-convolution square root of A264224.

Radius of convergence is r = 1/5 with A(1/5) = sqrt(5).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 144*x^5 + 582*x^6 + 2418*x^7 +...
where A(x) = A( -x/(1-4*x) ) / sqrt(1-4*x).
RELATED SERIES.
Note that A(x)^2 = A( x^2/(1-4*x) ) / sqrt(1-4*x):
A(x)^2 = 1 + 2*x + 7*x^2 + 26*x^3 + 103*x^4 + 422*x^5 + 1774*x^6 + 7604*x^7 + 33109*x^8 + 146042*x^9 + 651256*x^10 +...+ A264224(n-1)*x^n +...
where
A( x^2/(1-4*x) ) = 1 + x^2 + 4*x^3 + 19*x^4 + 88*x^5 + 410*x^6 + 1912*x^7 + 8933*x^8 + 41808*x^9 + 196016*x^10 + 920640*x^11 + 4331590*x^12 +...
Also,
A( x/(1+2*x) ) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 8*x^5 + 30*x^6 + 42*x^7 + 198*x^8 + 257*x^9 + 1385*x^10 + 1724*x^11 + 10072*x^12 +...
such that
A( x/(1+2*x) )^2 / (1+2*x) = 1 + 3*x^2 + 15*x^4 + 90*x^6 + 597*x^8 + 4212*x^10 + 30942*x^12 + 233766*x^14 + 1802706*x^16 +...
equals A( x^2/(1-4*x^2) ) / sqrt(1-4*x^2).
		

Crossrefs

Cf. A264224.

Programs

PARI

{a(n) = my(A=1+x,X=x+x*O(x^n)); for(i=1, n, A = subst(A, x, x^2/(1-4*X))^(1/2)/(1-4*X)^(1/4)  ); polcoeff(A, n)}
for(n=0,30, print1(a(n),", "))

Formula

G.f. A(x) satisfies:

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

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