A201970 G.f. satisfies: A(x) = 1 + x*A(x)^2 + x^2/A(x)^3.

1, 1, 3, 4, 11, 53, 146, 366, 1563, 5837, 16720, 59917, 246090, 828157, 2773984, 11056545, 41539923, 142602611, 536781500, 2103092222, 7632057148, 27948758125, 108960963441, 413531196001, 1523673835870, 5833675306068, 22596747340839, 85028853983908, 322415760345476

Offset

0,3

Comments

This sequence consists entirely of positive integers, but negative coefficients appear in G(x) where G(x) = 1 + x*G(x)^2 + x^2/G(x)^4.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Vaclav Kotesovec, Recurrence (of order 8)

Formula

G.f. satisfies: x^2 = A(x)^4 - A(x)^3 - x*A(x)^5.

a(n) ~ s*sqrt((2*r + s^5) / (Pi*(6*r + s^5))) / (2*n^(3/2)*r^n), where r = 0.2481615785281025222129020972009425192154180157568... and s = 2.036459583599244077761952153978454195241402482624... are roots of the system of equations 1 + r^2/s^3 + r*s^2 = s, 2*r*s = 1 + 3*r^2/s^4. - Vaclav Kotesovec, Nov 18 2017

Example

G.f.: A(x) = 1 + x + 3*x^2 + 4*x^3 + 11*x^4 + 53*x^5 + 146*x^6 +...
Related expansions begin:
A(x)^2 = 1 + 2*x + 7*x^2 + 14*x^3 + 39*x^4 + 152*x^5 + 480*x^6 +...
1/A(x)^3 = 1 - 3*x - 3*x^2 + 14*x^3 - 6*x^4 - 114*x^5 + 133*x^6 +...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A^2+x^2/(A+x*O(x^n))^3); polcoeff(A, n)}

Crossrefs

Sequence in context: A101982 A041947 A327081 * A102013 A192223 A079940

Adjacent sequences: A201967 A201968 A201969 * A201971 A201972 A201973

Keyword

nonn

Author

Paul D. Hanna, Dec 07 2011

Status

approved