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A129086 Coefficients of solution to A(x) = (1 + x*A(x)^2) * (1-3*x) / (1-2*x)^2. 0
1, 2, 5, 14, 42, 133, 443, 1552, 5716, 22068, 88830, 370209, 1585841, 6936459, 30813483, 138445492, 627256282, 2859652414, 13099023380, 60225071992, 277729496928, 1283986487874, 5948991719082, 27616185153765 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..23.

FORMULA

Hankel transform of a(n) is A006720(n+3).

G.f. A(x) satisfies 0 = f(x, A(x)) where f(u, v) = (v-1) + (3 - 4*v - v^2) * u + (4*v + 3*v^2) * u^2.

Let s(n)= A006769(n). Then 0 = f( s(n+4) * s(n+6) / ( s(n) * s(n+10)), -s(n) * s(n+7) / ( s(n+3) * s(n+4)) ) where f(u, v) = (v-1) + (3 - 4*v - v^2) * u + (4*v + 3*v^2) * u^2.

G.f.: ((1 - 2*x)^2 - sqrt((1 - 4*x) * (1 - 8*x + 16*x^2 - 4*x^3) )) / (2*x * (1 - 3*x)).

a(n) ~ sqrt((s^2 - 6*r - 3) / (Pi*(1 - 5*r + 6*r^2))) / (2*n^(3/2) * r^n), where r = 0.2039479457772143062225326263671960106786457685654... and s = 2.214319743377535187415497700848580488907919637219... are real roots of the system of equations 1 = r*s^2, 2*(1 - 3*r)*r*s = (1 - 2*r)^2. - Vaclav Kotesovec, Nov 27 2017

MATHEMATICA

CoefficientList[Series[((1 - 2*x)^2 - Sqrt[(1 - 4*x) * (1 - 8*x + 16*x^2 - 4*x^3) ]) / (2*x * (1 - 3*x)), {x, 0, 25}], x] (* Vaclav Kotesovec, Nov 27 2017 *)

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( ((1 - 2*x)^2 - sqrt((1 - 4*x) * (1 - 8*x + 16*x^2 - 4*x^3) + x^2 * O(x^n))) / (2*x * (1 - 3*x)), n))};

(PARI) {a(n) = my(A); if( n<0, 0, A = 1 + O(x); for(k= 1, n, A = (1 + x*A^2) * (1 - 3*x) / (1 - 2*x)^2 ); polcoeff(A, n))};

CROSSREFS

Sequence in context: A162748 A061815 A202061 * A035052 A148330 A149876

Adjacent sequences:  A129083 A129084 A129085 * A129087 A129088 A129089

KEYWORD

nonn

AUTHOR

Michael Somos, Mar 29 2007

STATUS

approved

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Last modified January 22 05:11 EST 2019. Contains 319353 sequences. (Running on oeis4.)