A337849 G.f.: 1 = Sum_{n>=0} a(n) * x^n / (1 + x*(1+x)^n)^(n+1).

1, 1, 1, 3, 9, 45, 237, 1591, 11795, 99651, 928507, 9474043, 104866399, 1249073623, 15914416345, 215724860511, 3097002496225, 46904398032017, 746831626559889, 12463977258581151, 217445976215654257, 3956098180940284169, 74897945785223884653

Offset: 0

Author

Paul D. Hanna, Sep 26 2020

Keywords

• nonn

Examples

			G.f.: 1 = 1/(1 + x) + x/(1 + x*(1+x))^2 + x^2/(1 + x*(1+x)^2)^3 + 3*x^3/(1 + x*(1+x)^3)^4 + 9*x^4/(1 + x*(1+x)^4)^5 + 45*x^5/(1 + x*(1+x)^5)^6 + 237*x^6/(1 + x*(1+x)^6)^7 + 1591*x^7/(1 + x*(1+x)^7)^8 + 11795*x^8/(1 + x*(1+x)^8)^9 + 99651*x^9/(1 + x*(1+x)^9)^10 + ... + a(n)*x^n/(1 + x*(1+x)^n)^(n+1) + ...
		

Links

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

Crossrefs

Cf. A177450.

Programs

PARI

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = -polcoeff( sum(m=0,#A-1, A[m+1] * x^m / (1 + x*(1+x)^m +O(x^#A) )^(m+1) ), #A-1) );A[n+1]}
\\ Print initial terms:
for(n=0,30,print1(a(n),", "))
\\ Verify sum equals unity:
sum(n=0,40, a(n)*x^n / (1 + x*(1+x)^n +O(x^41) )^(n+1) )

Formula

G.f.: 1 = Sum_{n>=0} a(n) * x^n / (1 + x*(1+x)^n)^(n+1).

G.f.: 1 = Sum_{n>=0} a(n) * x^n * (1-x)^(n^2+n+1) / ((1-x)^(n+1) + x)^(n+1).