A264414 G.f. A(x) satisfies: A(x)^2 = A(x^2) + 20*x.

1, 10, -45, 450, -5535, 75600, -1106100, 16953750, -268652880, 4365638550, -72354858300, 1218356280000, -20784495119850, 358457180010750, -6239532583193625, 109476057598087500, -1934128026918961515, 34378012275668994150, -614328464414815220025, 11030366153872043358750, -198899407327466712808800, 3600377821710426377668500

Offset: 0

Author

Paul D. Hanna, Nov 12 2015

Keywords

• sign

Examples

			G.f.: A(x) = 1 + 10*x - 45*x^2 + 450*x^3 - 5535*x^4 + 75600*x^5 - 1106100*x^6 +...
where
A(x)^2 = 1 + 20*x + 10*x^2 - 45*x^4 + 450*x^6 - 5535*x^8 + 75600*x^10 - 1106100*x^12 +...
so that A(x)^2 = A(x^2) + 20*x.
Let G(x) = Series_Reversion( x / (A(x^2) + 4*x) ), then
G(x) = x + 4*x^2 + 28*x^3 + 208*x^4 + 1702*x^5 + 14584*x^6 + 129808*x^7 + 1187008*x^8 + 11089153*x^9 + 105370660*x^10 +...+ A264226(n)*x^n +...
such that G(x)^2 = G( x^2/(1-8*x) ).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A271957, A264226, A264412, A264413, A264415.

Programs

PARI

{a(n) = my(A=1); for(i=1,n, A = sqrt( subst(A,x,x^2) + 20*x +x*O(x^n))); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

Given g.f. A(x), let G(x) denote the g.f. of A264226, then:

(1) G( x/(A(x)^2 - 16*x) ) = x,

(2) G( x/(A(x^2) + 4*x) ) = x,

(3) A(G(x))^2 = (1+16*x) * G(x)/x,

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

where G(x)^2 = G( x^2/(1-8*x) ).