A264412 G.f. A(x) satisfies: A(x)^2 = A(x^2) + 6*x.

1, 3, -3, 9, -33, 126, -513, 2214, -9876, 45045, -209493, 990198, -4741191, 22946247, -112079214, 551793303, -2735330190, 13641353118, -68394016548, 344539469889, -1743035351958, 8851923849123, -45110440515753, 230615809867476, -1182376529280117, 6078184963674498, -31322206517658453, 161774639164275552, -837290923919381322

Offset

0,2

Links

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

Formula

Given g.f. A(x), let G(x) denote the g.f. of A264224, then:

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

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

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

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

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

a(n) ~ c * (-1)^(n+1) * d^n / n^(3/2), where d = 5.46806882358680646837..., c = 0.268849330049069376... . - Vaclav Kotesovec, Nov 18 2015

Example

G.f.: A(x) = 1 + 3*x - 3*x^2 + 9*x^3 - 33*x^4 + 126*x^5 - 513*x^6 + 2214*x^7 - 9876*x^8 + 45045*x^9 +...
where
A(x)^2 = 1 + 6*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 +...
so that A(x)^2 = A(x^2) + 6*x.
Let G(x) = Series_Reversion( x / (A(x^2) + 2*x) ), then
G(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 +...+ A264224(n)*x^n +...
such that G(x)^2 = G( x^2/(1-4*x) ) and A(G(x))^2 = (1+4*x) * G(x)/x.

Prog

(PARI) {a(n) = my(A=1); for(i=1, n, A = sqrt( subst(A, x, x^2) + 6*x +x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A271930, A264224, A264413, A264414, A264415.

Sequence in context: A327712 A257611 A268617 * A202889 A257620 A032086

Adjacent sequences: A264409 A264410 A264411 * A264413 A264414 A264415

Keyword

sign

Author

Paul D. Hanna, Nov 12 2015

Status

approved