A323573 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n - 1)^n/(A(x) + (1+x)^n - 1)^(n+1).

1, 1, 1, 6, 41, 381, 4387, 59961, 946119, 16901219, 336924252, 7412401676, 178372705409, 4660680881897, 131410732869312, 3977081948965664, 128600945014475040, 4424941538152614645, 161433547224627797940, 6224586371820817112652, 252934418382142622780667, 10803348636926511625239387, 483881915960470248201012949

Offset: 0

Author

Paul D. Hanna, Feb 07 2019

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + x^2 + 6*x^3 + 41*x^4 + 381*x^5 + 4387*x^6 + 59961*x^7 + 946119*x^8 + 16901219*x^9 + 336924252*x^10 + 7412401676*x^11 + 178372705409*x^12 + ...
such that
1 = 1/A(x) +  x/(A(x) + x)^2 + ((1+x)^2 - 1)^2/(A(x) + (1+x)^2 - 1)^3 + ((1+x)^3 - 1)^3/(A(x) + (1+x)^3 - 1)^4 + ((1+x)^4 - 1)^4/(A(x) + (1+x)^4 - 1)^5 + ((1+x)^5 - 1)^5/(A(x) + (1+x)^5 - 1)^6 + ((1+x)^2 - 1)^6/(A(x) + (1+x)^6 - 1)^7 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A323313.

Programs

PARI

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

Formula

a(n) ~ c * n^n / (exp(n) * log(2)^(2*n)), where c = 0.51205951699411... - Vaclav Kotesovec, Aug 11 2021