A323313 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n - 1)^n/(A(x) + 1 - (1+x)^n)^(n+1).

1, 1, 5, 42, 553, 9757, 213989, 5577285, 167819725, 5715066723, 217100774130, 9097820151376, 416870148834739, 20734141087348929, 1112567525238708108, 64067000804518722936, 3941130502347547574782, 257950869129924123671281, 17899157911139617070534480, 1312572773247815201019870104, 101429898156047011931583837037, 8238300763728888616466106146271, 701648737726537575830481016280283

Offset: 0

Author

Paul D. Hanna, Feb 02 2019

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 42*x^3 + 553*x^4 + 9757*x^5 + 213989*x^6 + 5577285*x^7 + 167819725*x^8 + 5715066723*x^9 + 217100774130*x^10 + ...
such that
1 = 1/A(x) + ((1+x) - 1)/(A(x) + 1 - (1+x))^2  +  ((1+x)^2 - 1)^2/(A(x) + 1 - (1+x)^2)^3  +  ((1+x)^3 - 1)^3/(A(x) + 1 - (1+x)^3)^4  +  ((1+x)^4 - 1)^4/(A(x) + 1 - (1+x)^4)^5  +  ((1+x)^5 - 1)^5/(A(x) + 1 - (1+x)^5)^6 + ...
also,
1 = 1/(A(x) + 2)  +  (1 + (1+x))/(A(x) + 1 + (1+x))^2  +  (1 + (1+x)^2)^2/(A(x) + 1 + (1+x)^2)^3  +  (1 + (1+x)^3)^3/(A(x) + 1 + (1+x)^3)^4  +  (1 + (1+x)^4)^4/(A(x) + 1 + (1+x)^4)^5  +  (1 + (1+x)^5)^5/(A(x) + 1 + (1+x)^5)^6 + ...
		

Links

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

Crossrefs

Cf. A323311, A323573.

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

G.f. A(x) satisfies:

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

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

a(n) ~ c * A317904^n * n^n / exp(n), where c = 0.546536986945124389069... - Vaclav Kotesovec, Aug 11 2021