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 A323686 G.f.: Sum_{n>=0} x^n * ((1+x)^n + (-1)^n)^n / (1 - (-1)^n*x*(1+x)^n)^(n+1). 1

%I

%S 1,1,6,19,98,483,2713,16183,102982,694083,4922791,36609764,284389271,

%T 2300588164,19330419610,168313262055,1515639311976,14089546025807,

%U 134997931892389,1331256937687764,13493866107417156,140422597267156783,1498630308004089329,16386210045999041610,183397194510164874411,2099282060320291913938,24556888617117856139092

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

%H Paul D. Hanna, <a href="/A323686/b323686.txt">Table of n, a(n) for n = 0..500</a>

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

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

%e G.f.: A(x) = 1 + x + 6*x^2 + 19*x^3 + 98*x^4 + 483*x^5 + 2713*x^6 + 16183*x^7 + 102982*x^8 + 694083*x^9 + 4922791*x^10 + 36609764*x^11 + ...

%e such that

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

%e also,

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

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

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

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

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

%Y Cf. A323680.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 12 2019

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Last modified September 25 03:11 EDT 2021. Contains 347651 sequences. (Running on oeis4.)