A292927 G.f. A(x) satisfies: A( x^2*A(x) - x*A(x)^3 ) = x^3.

1, 1, 3, 11, 52, 258, 1343, 7257, 40275, 228278, 1315922, 7691196, 45473095, 271482064, 1634359974, 9910367591, 60474714189, 371087272878, 2288372703482, 14174212020218, 88145412404781, 550128210470715, 3444680265887877, 21633697884627237, 136238869051956545, 860130515526195618, 5442975808350831237, 34517730741744461395, 219338548934112758627

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..520

Formula

a(n) ~ c * d^n / n^(3/2), where d = 6.705143079646414499260567437823218217... and c = 0.03594678018676382296451433... - Vaclav Kotesovec, Oct 10 2017

Example

G.f.: A(x) = x + x^2 + 3*x^3 + 11*x^4 + 52*x^5 + 258*x^6 + 1343*x^7 + 7257*x^8 + 40275*x^9 + 228278*x^10 + 1315922*x^11 + 7691196*x^12 + 45473095*x^13 + 271482064*x^14 + 1634359974*x^15 + 9910367591*x^16 +...
such that A( x^2*A(x) - x*A(x)^3 ) = x^3.
RELATED SERIES.
x^2*A(x) - x*A(x)^3 = x^3 - x^6 - x^9 - x^12 - 8*x^15 - 13*x^18 - 37*x^21 - 159*x^24 - 388*x^27 - 1403*x^30 - 5090*x^33 - 15931*x^36 - 58532*x^39 +...
Let B(x) be the series reversion of A(x), so that B(A(x)) = x, then
B(x) = x - x^2 - x^3 - x^4 - 8*x^5 - 13*x^6 - 37*x^7 - 159*x^8 - 388*x^9 - 1403*x^10 - 5090*x^11 - 15931*x^12 - 58532*x^13 - 207536*x^14 - 719812*x^15 - 2641077*x^16 - 9504900*x^17 - 34393816*x^18 - 126750932*x^19 - 464389638*x^20 +...
then x^2*A(x) - x*A(x)^3 = B(x^3).

Prog

(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -Vec(subst(F, x, x^2*F - x*F^3))[#A] ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Crossrefs

Cf. A265940, A268039, A272463, A292928.

Sequence in context: A129097 A319155 A362468 * A367046 A179322 A014510

Adjacent sequences: A292924 A292925 A292926 * A292928 A292929 A292930

Keyword

nonn

Author

Paul D. Hanna, Sep 26 2017

Status

approved