A231617 G.f. satisfies: A(x) = (1 - 3*x*A(x)^2) * sqrt(4*A(x)^2 - 3).

1, 1, 8, 75, 788, 8914, 106006, 1306629, 16544772, 213925368, 2812797588, 37494368574, 505536154470, 6882295486576, 94473351706766, 1306171811733083, 18172571198392164, 254235687592867548, 3574318400418780952, 50473259265229118344, 715565619086065023572, 10181073360665458354752

Offset

0,3

Comments

Self-convolution square yields A231616.

Links

G. C. Greubel, Table of n, a(n) for n = 0..845

Formula

G.f. A(x) satisfies:

(1) A(x) = exp( x*(2*A(x)^4 - 3*A(x)^2/2) + Integral(2*A(x)^4 - 3*A(x)^2/2 dx) ).

(2) A(x) = sqrt( (1/x)*Series_Reversion( x*(1-2*x)*(1-6*x)/(1-3*x)^2 ) ).

(3) A(x) = sqrt( 1 + x*A(x)^2*(4*A(x)^2 - 3)*(2 - 3*x*A(x)^2) ).

(4) A(x) = sqrt( 1 + Sum_{n>=2} 3^(n-2) * n * x^(n-1) * A(x)^(2*n) ).

D-finite with recurrence: 4*n*(2*n-1)*(2*n+1)*(109*n^3 - 570*n^2 + 959*n - 522)*a(n) = 8*(2*n-1)*(1853*n^5 - 11543*n^4 + 26548*n^3 - 27922*n^2 + 13188*n - 2160)*a(n-1) - 9*(n-2)*(5341*n^5 - 33271*n^4 + 77057*n^3 - 81281*n^2 + 37626*n - 5760)*a(n-2) + 162*(n-3)*(n-2)*(2*n-5)*(109*n^3 - 243*n^2 + 146*n - 24)*a(n-3). - Vaclav Kotesovec, Dec 20 2013

a(n) ~ c * r^n / (sqrt(Pi)*n^(3/2)), where r = 1/48*(272 + (12606848 - 188352*sqrt(327))^(1/3) + 4*(196982 + 2943*sqrt(327))^(1/3)) = 15.283249955997317489... is the root of the equation 441*r - 272*r^2 + 16*r^3 = 324, and c = 1/2616*sqrt(327)*sqrt((164658779 + 6502068 * sqrt(327))^(1/3)*((164658779 + 6502068 * sqrt(327))^(2/3) + 236857 + 218*(164658779 + 6502068 * sqrt(327))^(1/3)))/((164658779 + 6502068*sqrt(327))^(1/3)) = 0.242927508847491211... is the root of the equation 446464*c^6 - 13952*c^4 -579*c^2 = 9. - Vaclav Kotesovec, Dec 20 2013

Example

G.f.: A(x) = 1 + x + 8*x^2 + 75*x^3 + 788*x^4 + 8914*x^5 + 106006*x^6 +...
Related expansions.
1 - 3*x*A(x)^2 = 1 - 3*x - 6*x^2 - 51*x^3 - 498*x^4 - 5370*x^5 -...
sqrt(4*A(x)^2 - 3) = 1 + 4*x + 26*x^2 + 228*x^3 + 2330*x^4 + 25960*x^5 +...
4*A(x)^4 - 3*A(x)^2 = 1 + 10*x + 101*x^2 + 1102*x^3 + 12762*x^4 +...
log(A(x)) = x + 15*x^2/2 + 202*x^3/3 + 2755*x^4/4 + 38286*x^5/5 +...

Mathematica

CoefficientList[Sqrt[1/x*InverseSeries[Series[x*(1-2*x)*(1-6*x)/(1-3*x)^2, {x, 0, 20}], x]], x] (* Vaclav Kotesovec, Dec 20 2013 *)

Prog

(PARI) {a(n)=polcoeff(sqrt(serreverse(x*(1-2*x)*(1-6*x)/(1-3*x)^2 +x^2*O(x^n))/x), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); for(i=1, n, A=exp(x*(2*A^4-3*A^2/2)+intformal(2*A^4-3*A^2/2 +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); for(i=1, n, A=sqrt(1+x*A^2*(4*A^2-3)*(2-3*x*A^2) +x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A231616, A228966, A231552, A231553, A231554, A231556, A231615.

Sequence in context: A015578 A145600 A268085 * A094735 A067306 A361528

Adjacent sequences: A231614 A231615 A231616 * A231618 A231619 A231620

Keyword

nonn

Author

Paul D. Hanna, Nov 11 2013

Status

approved