A317340 G.f. A(x) satisfies: 1+x = Sum_{n>=0} ((1+x)^(n+1) - A(x))^n.

1, 1, 5, 19, 215, 3221, 60255, 1328529, 33525557, 949932155, 29820028655, 1026798211345, 38478492170097, 1559272992752559, 67959179777259427, 3170787800382000429, 157723858411070076821, 8334021579932292731707, 466249346920348934320021, 27536331208734157654465035, 1712202839967112789793314087, 111816679758439520925160452089

Offset: 0

Author

Paul D. Hanna, Aug 02 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 19*x^3 + 215*x^4 + 3221*x^5 + 60255*x^6 + 1328529*x^7 + 33525557*x^8 + 949932155*x^9 + 29820028655*x^10 + ...
such that
1+x = 1  +  ((1+x)^2 - A(x))  +  ((1+x)^3 - A(x))^2  +  ((1+x)^4 - A(x))^3  +  ((1+x)^5 - A(x))^4  +  ((1+x)^6 - A(x))^5  +  ((1+x)^7 - A(x))^6  +  ((1+x)^8 - A(x))^7 + ...
Also,
1+x = 1/(1 + A(x))  +  (1+x)^2/(1 + (1+x)*A(x))^2  +  (1+x)^6/(1 + (1+x)^2*A(x))^3  +  (1+x)^12/(1 + (1+x)^3*A(x))^4  +  (1+x)^20/(1 + (1+x)^4*A(x))^5  +  (1+x)^30/(1 + (1+x)^5*A(x))^6  +  (1+x)^42/(1 + (1+x)^6*A(x))^7 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A304642, A303056.

Programs

PARI

{a(n) = my(A=[1,1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(m+1) - Ser(A))^m ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:

(1) 1+x = Sum_{n>=0} ((1+x)^(n+1) - A(x))^n.

(2) 1+x = Sum_{n>=0} (1+x)^(n*(n+1)) / (1 + (1+x)^n*A(x))^(n+1).

a(n) ~ c * r^(2*n) * (1 + exp(1/r))^n * n^n / exp(n), where r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation (1 + exp(-1/r)) * LambertW(-exp(-1/r)/r) = -1/r, and c = 0.9243009841585606... - Vaclav Kotesovec, Aug 06 2018