A322618 - OEIS

%I #11 Jan 26 2019 10:31:56

%S 1,-1,4,-11,24,-47,95,-204,440,-915,1829,-3556,6819,-13043,25039,

%T -48306,93509,-181219,351089,-679814,1316526,-2552828,4961602,

%U -9672267,18916895,-37114623,73027149,-144047576,284741852,-563872928,1118354547,-2221007241,4415827590,-8788137259,17504252965,-34889806960,69584723922,-138850448656,277179930085,-553510258815,1105635941562

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

%H Paul D. Hanna, <a href="/A322618/b322618.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%e G.f.: A(x) = 1 - x + 4*x^2 - 11*x^3 + 24*x^4 - 47*x^5 + 95*x^6 - 204*x^7 + 440*x^8 - 915*x^9 + 1829*x^10 - 3556*x^11 + 6819*x^12 - 13043*x^13 + 25039*x^14 + ...

%e such that

%e A(x) = 1/(1 + 2*x) + x*(1+x)/(1+x + x^2)^2 + x^2*(1+x^2)^2/(1+x + x^3)^3 + x^3*(1+x^3)^3/(1+x + x^4)^4 + x^4*(1+x^4)^4/(1+x + x^5)^5 + x^5*(1+x^5)^5/(1+x + x^6)^6 + ...

%e also,

%e A(x) = 1 - x*(1-x)/(1+x - x^2)^2 + x^2*(1-x^2)^2/(1+x - x^3)^3 - x^3*(1-x^3)^3/(1+x - x^4)^4 + x^4*(1-x^4)^4/(1+x - x^5)^5 - x^5*(1-x^5)^5/(1+x - x^6)^6 + ...

%o (PARI) {a(n) = my(A = sum(m=0,n, x^m*(1 + x^m)^m/(1 + x + x^(m+1) +x*O(x^n) )^(m+1) ) ); polcoeff(A,n)}

%o for(n=0,50, print1(a(n),", "))

%o (PARI) {a(n) = my(A = sum(m=0,n, (-x)^m*(1 - x^m)^m/(1 + x - x^(m+1) +x*O(x^n) )^(m+1) ) ); polcoeff(A,n)}

%o for(n=0,50, print1(a(n),", "))

%Y Cf. A322619.

%K sign

%O 0,3

%A _Paul D. Hanna_, Jan 26 2019