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A231556 G.f. satisfies: A(x) = (1 - x*A(x))^2 * (2*A(x) - 1). 9
1, 2, 11, 74, 556, 4472, 37667, 328010, 2929230, 26679916, 246889782, 2314629836, 21937826010, 209856958820, 2023513559779, 19646643605914, 191911292243206, 1884679009555852, 18597046827999018, 184291418008942092, 1833316710122314192, 18301376717308102560 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..800

FORMULA

G.f. A(x) satisfies:

(1) A(x) = exp( x*(2*A(x)^2 - A(x)) + Integral(2*A(x)^2 - A(x) dx) ).

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

(3) A(x) = 1 + x*A(x)*(2*A(x) - 1)*(2 - x*A(x)).

(4) A(x) = 1 + Sum_{n>=2} n * x^(n-1) * A(x)^n.

Recurrence: 8*n*(n+1)*(55*n-69)*a(n) = 4*n*(1210*n^2 - 2123*n + 771)*a(n-1) - (1375*n^3 - 4475*n^2 + 4146*n - 1008)*a(n-2) + 2*(n-3)*(2*n-3)*(55*n-14)*a(n-3). - Vaclav Kotesovec, Dec 21 2013

a(n) ~ 1/1320*sqrt(330)*sqrt((1310475375 + 5626995*sqrt(330))^(1/3)*((1310475375 + 5626995*sqrt(330))^(2/3) + 1195095 + 990*(1310475375 + 5626995*sqrt(330))^(1/3))) / ((1310475375 + 5626995*sqrt(330))^(1/3) * sqrt(Pi)) * 12^(-n)*((75716 + 330*sqrt(330))^(2/3) + 1786 + 44*(75716 + 330*sqrt(330))^(1/3))^n * (75716 + 330*sqrt(330))^(-n/3) * (1/n)^(3/2). - Vaclav Kotesovec, Dec 21 2013

EXAMPLE

G.f.: A(x) = 1 + 2*x + 11*x^2 + 74*x^3 + 556*x^4 + 4472*x^5 +...

Related expansions.

(1 - x*A(x))^2 = 1 - 2*x - 3*x^2 - 18*x^3 - 122*x^4 - 920*x^5 -...

2*A(x) - 1 = 1 + 4*x + 22*x^2 + 148*x^3 + 1112*x^4 + 8944*x^5 +...

2*A(x)^2 - A(x) = 1 + 6*x + 41*x^2 + 310*x^3 + 2502*x^4 + 21120*x^5 +...

log(A(x)) = 2*x + 18*x^2/2 + 164*x^3/3 + 1550*x^4/4 + 15012*x^5/5 +...

MATHEMATICA

CoefficientList[InverseSeries[Series[x*(1-4*x+2*x^2)/(1-x)^2, {x, 0, 20}], x]/x, x] (* Vaclav Kotesovec, Dec 21 2013 *)

PROG

(PARI) {a(n)=polcoeff((serreverse(x*(1-4*x+2*x^2)/(1-x)^2 +x^2*O(x^n))/x), n)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n)=local(A=1); for(i=1, n, A=exp(x*(2*A^2-A)+intformal(2*A^2-A +x*O(x^n)))); polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n)=local(A=1); for(i=1, n, A=1+x*A*(2*A-1)*(2-x*A) +x*O(x^n)); polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A228966, A231552, A231553, A231554, A231615, A231616, A231618.

Sequence in context: A299786 A199417 A114179 * A207397 A319743 A166992

Adjacent sequences:  A231553 A231554 A231555 * A231557 A231558 A231559

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 10 2013

STATUS

approved

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Last modified October 20 05:42 EDT 2019. Contains 328247 sequences. (Running on oeis4.)