A301931 G.f. A(x) satisfies: A(x) = x*(1 + A(x)*A'(x)) / (1 - A(x)*A'(x)).

1, 2, 14, 154, 2186, 36930, 712158, 15295602, 360155378, 9196938274, 252714360398, 7428201595178, 232480106673562, 7717715629942274, 270896581615492926, 10025955123898058082, 390290027013023089122, 15945008348985522925890, 682270367930391378957198, 30518971384549282782489786, 1424657384555111198491140138, 69290703924209121823629630018

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 + 14*x^3 + 154*x^4 + 2186*x^5 + 36930*x^6 + 712158*x^7 + 15295602*x^8 + 360155378*x^9 + 9196938274*x^10 + ...
RELATED SERIES.
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 + ...
		

Links

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

Crossrefs

Cf. A301930, A301932, A301933.

Programs

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)}
for(n=1,30,print1(a(n),", "))

Formula

a(n) ~ c * 2^n * n! * n^2, where c = 0.03425366633935068788883129... - Vaclav Kotesovec, Oct 14 2020