A301932 G.f. A(x) satisfies: A(x) = x*(1 + 3*A(x)*A'(x)) / (1 + A(x)*A'(x)).

1, 2, 10, 74, 710, 8322, 115018, 1828962, 32852526, 657188258, 14477811178, 348100068698, 9067809569750, 254354791759298, 7642986480897930, 244923580410697938, 8337728465913016926, 300482221889444825154, 11429089791630856291018, 457542303069698601849194, 19230862148761320966737254, 846710680545018639230252418

Offset: 1

Author

Paul D. Hanna, Mar 28 2018

Keywords

• nonn

Comments

Compare to: C(x) = x*(1 + 2*C(x)*C'(x)) / (1 + C(x)*C'(x)) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = x + 2*x^2 + 10*x^3 + 74*x^4 + 710*x^5 + 8322*x^6 + 115018*x^7 + 1828962*x^8 + 32852526*x^9 + 657188258*x^10 + ...
such that A = A(x) satisfies: A = x*(1 + 3*A*A')/(1 + A*A').
RELATED SERIES.
A(x)*A'(x) = x + 6*x^2 + 48*x^3 + 470*x^4 + 5448*x^5 + 73374*x^6 + 1132000*x^7 + 19752822*x^8 + 385285080*x^9 + 8311631702*x^10 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..400

Crossrefs

Cf. A301930, A301931, A301933.

Programs

PARI

{a(n) = my(L=x); for(i=1,n, L = x*(1 + 3*L'*L)/(1 + L'*L +x*O(x^n)) ); polcoeff(L,n)}
for(n=1,30,print1(a(n),", "))

Formula

a(n) ~ c * 2^n * n!, where c = 0.181799839377767875340143846... - Vaclav Kotesovec, Oct 14 2020