login
G.f. A(x) satisfies: A(x) = x*(1 + A(x)*A'(x)) / (1 - A(x)*A'(x)).
4

%I #9 Oct 14 2020 08:36:18

%S 1,2,14,154,2186,36930,712158,15295602,360155378,9196938274,

%T 252714360398,7428201595178,232480106673562,7717715629942274,

%U 270896581615492926,10025955123898058082,390290027013023089122,15945008348985522925890,682270367930391378957198,30518971384549282782489786,1424657384555111198491140138,69290703924209121823629630018

%N G.f. A(x) satisfies: A(x) = x*(1 + A(x)*A'(x)) / (1 - A(x)*A'(x)).

%C 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).

%H Paul D. Hanna, <a href="/A301931/b301931.txt">Table of n, a(n) for n = 1..400</a>

%F a(n) ~ c * 2^n * n! * n^2, where c = 0.03425366633935068788883129... - _Vaclav Kotesovec_, Oct 14 2020

%e G.f.: A(x) = x + 2*x^2 + 14*x^3 + 154*x^4 + 2186*x^5 + 36930*x^6 + 712158*x^7 + 15295602*x^8 + 360155378*x^9 + 9196938274*x^10 + ...

%e RELATED SERIES.

%e A(x)*A'(x) = x + 6*x^2 + 64*x^3 + 910*x^4 + 15552*x^5 + 304206*x^6 + 6627840*x^7 + 158162238*x^8 + 4087933120*x^9 + 113539676470*x^10 + ...

%o (PARI) {a(n) = my(L=x); for(i=1,n, L = x*(1 + L'*L)/(1 - L'*L +x*O(x^n)) ); polcoeff(L,n)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A301930, A301932, A301933.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Mar 28 2018