A264413 G.f. A(x) satisfies: A(x)^2 = A(x^2) + 12*x.

1, 6, -15, 90, -660, 5310, -45765, 413640, -3864345, 37014120, -361577790, 3588484140, -36079979085, 366728363460, -3762120325140, 38901621985290, -405039437707575, 4242802537386450, -44681704461745740, 472795814216587140, -5024232597805717410, 53596341229925979360, -573736849659978481665, 6161218734911098973490, -66355728143871653462745

Offset: 0

Author

Paul D. Hanna, Nov 12 2015

Keywords

• sign

Examples

			G.f.: A(x) = 1 + 6*x - 15*x^2 + 90*x^3 - 660*x^4 + 5310*x^5 - 45765*x^6 + 413640*x^7 - 3864345*x^8 + 37014120*x^9 - 361577790*x^10 +...
where
A(x)^2 = 1 + 12*x + 6*x^2 - 15*x^4 + 90*x^6 - 660*x^8 + 5310*x^10 - 45765*x^12 + 413640*x^14 - 3864345*x^16 + 37014120*x^18 - 361577790*x^20 +...
so that A(x)^2 = A(x^2) + 12*x.
Let G(x) = Series_Reversion( x / (A(x^2) + 3*x) ), then
G(x) = x + 3*x^2 + 15*x^3 + 81*x^4 + 462*x^5 + 2718*x^6 + 16344*x^7 + 99900*x^8 + 618567*x^9 + 3870909*x^10 +...+ A264225(n)*x^n +...
such that G(x)^2 = G( x^2/(1-6*x) ).
		

Links

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

Crossrefs

Cf. A271935, A264225, A264233, A264412, A264414, A264415.

Programs

PARI

{a(n) = my(A=1); for(i=1,n, A = sqrt( subst(A,x,x^2) + 12*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 A264225, then:

(1) G( x/(A(x)^2 - 9*x) ) = x,

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

(3) A(G(x))^2 = (1+9*x) * G(x)/x,

(4) A(G(x)^2) = (1-3*x) * G(x)/x,

where G(x)^2 = G( x^2/(1-6*x) ).