A295812 G.f. A(x) satisfies: G(A(x)) = exp(x), where G(x) equals the e.g.f. of A296170.

1, 1, 3, 19, 226, 4259, 110514, 3626207, 143043592, 6567931068, 343278693103, 20092744961109, 1300754163383700, 92223505422990050, 7104166647498916816, 590661172651143976231, 52710327177111760030280, 5024720072707894279118236, 509553454073135435969780828, 54771493019290133717304608756, 6220332385328132888848047735930, 744260531662484056612631555859467

Offset: 1

Author

Paul D. Hanna, Dec 09 2017

Keywords

• nonn

Comments

E.g.f. G(x) of A296170 satisfies: [x^(n-1)] G(x)^(n^2) = [x^n] G(x)^(n^2) for n>=1.

Examples

			G.f. A(x) = x + x^2 + 3*x^3 + 19*x^4 + 226*x^5 + 4259*x^6 + 110514*x^7 + 3626207*x^8 + 143043592*x^9 + 6567931068*x^10 + 343278693103*x^11 + 20092744961109*x^12 + 1300754163383700*x^13 + 92223505422990050*x^14 + 7104166647498916816*x^15 +...
The series reversion equals the logarithm of the e.g.f. of A296170, which begins:
Series_Reversion(A(x)) = x - x^2 - x^3 - 9*x^4 - 134*x^5 - 2852*x^6 - 79096*x^7 - 2699480*x^8 - 109201844*x^9 - 5100872244*x^10 - 269903909820*x^11 - 15944040740604*x^12 - 1039553309158964*x^13 - 74123498185170292*x^14 - 5736368141560365292*x^15 +...+ A296171(n)*x^n +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A296170, A296171, A295813, A295814.

Programs

PARI

{a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^2)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^2 ); polcoeff(serreverse(log(Ser(A))),n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. is the series reversion of the logarithm of the e.g.f. of A296170.

a(n) ~ c * d^n * n! / n^3, where d = -4/(LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = (2 + LambertW(-2*exp(-2)))^2 * sqrt(-LambertW(-2*exp(-2))*(1 + LambertW(-2*exp(-2)))) / (8*sqrt(2)*Pi) = 0.0350943105... - Vaclav Kotesovec, Dec 22 2017, updated Aug 06 2018