A264224 G.f. A(x) satisfies: A(x)^2 = A( x^2/(1-4*x) ), with A(0) = 0.

1, 2, 7, 26, 103, 422, 1774, 7604, 33109, 146042, 651256, 2931392, 13301038, 60775340, 279393742, 1291311620, 5996491666, 27962898020, 130883946751, 614664907706, 2895279687655, 13674609742598, 64744203198388, 307221794213768, 1460778188820220, 6958635514922552, 33205258829750809, 158699556581760134

Offset: 1

Author

Paul D. Hanna, Nov 08 2015

Keywords

• nonn

Comments

Radius of convergence is r = 1/5, where r = r^2/(1-4*r), with A(r) = 1.

Compare to a g.f. M(x) of Motzkin numbers: M(x)^2 = M(x^2/(1-2*x)) where M(x) = (1-x - sqrt(1-2*x-3*x^2))/(2*x).

Examples

			G.f.: A(x) = x + 2*x^2 + 7*x^3 + 26*x^4 + 103*x^5 + 422*x^6 + 1774*x^7 + 7604*x^8 + 33109*x^9 + 146042*x^10 + 651256*x^11 + 2931392*x^12 +...
where A(x)^2 = A(x^2/(1-4*x)).
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 18*x^4 + 80*x^5 + 359*x^6 + 1620*x^7 + 7354*x^8 + 33568*x^9 + 154023*x^10 + 710156*x^11 + 3289142*x^12 + 15297744*x^13 +...
sqrt(A(x)/x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 144*x^5 + 582*x^6 + 2418*x^7 + 10266*x^8 + 44353*x^9 + 194395*x^10 +...+ A264231(n)*x^n +...
A( x/(1+2*x) ) = x + 3*x^3 + 15*x^5 + 90*x^7 + 597*x^9 + 4212*x^11 + 30942*x^13 + 233766*x^15 + 1802706*x^17 + 14122359*x^19 + 112033791*x^21 + 898024320*x^23 +...
A( x^2/(1-4*x^2) ) = x^2 + 6*x^4 + 39*x^6 + 270*x^8 + 1959*x^10 + 14706*x^12 + 113166*x^14 + 887004*x^16 + 7050837*x^18 + 56672622*x^20 + 459646488*x^22 +...
where A( x^2/(1-4*x^2) ) = A( x/(1+2*x) )^2.
Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where
B(x) = 1 + 2*x + 3*x^2 - 3*x^4 + 9*x^6 - 33*x^8 + 126*x^10 - 513*x^12 + 2214*x^14 - 9876*x^16 + 45045*x^18 - 209493*x^20 +...+ A264412(n)*x^(2*n) +...
such that B(x) = F(x^2) + 2*x = F(x)^2 - 4*x and F(x) is the g.f. of A264412.
PARTICULAR VALUES.
A(1/5) = 1.
A(-1/5) = -A(1/9) = -0.15262256991492310976978497600904...
A(1/6)^2 = A(1/12) = 0.10315964246752710052686298695713...
A(1/6)^4 = A(1/96) = 0.01064191183402802084987998396215...
A(1/7)^2 = A(1/21) = 0.053075120978549663441827849989065...
A(1/7)^4 = A(1/357) = 0.002816968466887682583828696137137...
A(1/8)^2 = A(1/32) = 0.033445065874191867268119916059631...
A(1/8)^4 = A(1/896) = 0.001118572431329033410718706838540...
A(1/9)^2 = A(1/45) = 0.0232936488474355927381514600230212...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A264231, A264232, A264412, A264225, A264226, A264227.

Programs

PARI

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

Formula

G.f. also satisfies:

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

(2) A( x/(1+2*x) ) = -A( -x/(1-2*x) ), an odd function.

(3) A( x/(1+2*x) )^2 = A( x^2/(1-4*x^2) ), an even function.

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

(5) [x^(2*n+1)] (x/A(x))^(2*n) = 0 for n>=0.

(6) [x^(2^n+k)] (x/A(x))^(2^n) = 0 for k=1..2^n-1, n>=1.

Given g.f. A(x), let F(x) denote the g.f. of A264412, then:

(7) A(x) = F(A(x))^2 * x/(1+4*x),

(8) A(x) = F(A(x)^2) * x/(1-2*x),

(9) A( x/(F(x)^2 - 4*x) ) = x,

(10) A( x/(F(x^2) + 2*x) ) = x,

where F(x)^2 = F(x^2) + 6*x.

Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * a(k+1) = 0 for odd n.

Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * a(k+1) = (-1)^n * a(n+1) for n>=0.

Sum_{k=0..n} binomial(n,k) * (+4)^(n-k) * a(k+1) = A264232(n+1) for n>=0.

Sum_{k=0..n} binomial(n,k) * (-8)^(n-k) * a(k+1) = (-1)^n * A264232(n+1) for n>=0.