A337849 - OEIS

%I #8 Oct 06 2020 19:24:24

%S 1,1,1,3,9,45,237,1591,11795,99651,928507,9474043,104866399,

%T 1249073623,15914416345,215724860511,3097002496225,46904398032017,

%U 746831626559889,12463977258581151,217445976215654257,3956098180940284169,74897945785223884653

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

%H Paul D. Hanna, <a href="/A337849/b337849.txt">Table of n, a(n) for n = 0..200</a>

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

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

%e 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) + ...

%o (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]}

%o \\ Print initial terms:

%o for(n=0,30,print1(a(n),", "))

%o \\ Verify sum equals unity:

%o sum(n=0,40, a(n)*x^n / (1 + x*(1+x)^n +O(x^41) )^(n+1) )

%Y Cf. A177450.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Sep 26 2020