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

%I #8 Mar 12 2016 11:22:00

%S 1,0,0,0,2,0,6,6,11,16,38,58,103,152,267,486,861,1392,2276,3812,6562,

%T 11224,18906,31446,52392,87776,147423,246698,411358,685988,1145964,

%U 1913208,3184968,5288788,8779115,14587396,24250424,40280516,66808605,110689870,183321646,303558816,502460050,831201108,1374306023,2271447536,3753011555,6198348494,10232046899,16883378468,27848975062,45923355280

%N G.f.: Sum_{n=-oo..+oo} ( x^n / (1 - x^(2*n)*(1+x)^(n+1)) )^2.

%H Paul D. Hanna, <a href="/A268656/b268656.txt">Table of n, a(n) for n = -2..400</a>

%F G.f.: [ Sum_{n=-oo..+oo} x^n / (1 - x^(2*n)*(1+x)^(n+1)) ]^2.

%F a(n) ~ ((1+sqrt(5))/2)^n * n / 20. - _Vaclav Kotesovec_, Mar 10 2016

%e G.f.: A(x) = 1/x^2 + 2*x^2 + 6*x^4 + 6*x^5 + 11*x^6 + 16*x^7 + 38*x^8 + 58*x^9 + 103*x^10 + 152*x^11 + 267*x^12 + 486*x^13 + 861*x^14 + 1392*x^15 + 2276*x^16 +...

%e such that the g.f. equals the sum of the squares

%e A(x) = Sum_{n=-oo..+oo} [ x^n / (1 - x^(2*n)*(1+x)^(n+1)) ]^2

%e and also the g.f. equals the square of the sum

%e A(x) = [ Sum_{n=-oo..+oo} x^n / (1 - x^(2*n)*(1+x)^(n+1)) ]^2

%e Thus

%e A(x)^(1/2) = Sum_{n=-oo..+oo} x^n / (1 - x^(2*n)*(1+x)^(n+1))

%e explicitly,

%e A(x)^(1/2) = 1/x + x^3 + 3*x^5 + 3*x^6 + 5*x^7 + 8*x^8 + 16*x^9 + 26*x^10 + 42*x^11 + 59*x^12 + 98*x^13 + 178*x^14 + 304*x^15 + 471*x^16 + 724*x^17 + 1167*x^18 + 1960*x^19 + 3266*x^20 + 5311*x^21 + 8521*x^22 + 13723*x^23 + 22288*x^24 + 36283*x^25 + 58825*x^26 + 95187*x^27 + 154542*x^28 + 251751*x^29 + 409437*x^30 +...

%o (PARI) {a(n) = my(A); A = sum(m=-2*n-4,n+2, ( x^m / (1 - x^(2*m)*(1+x)^(m+1)) +x^3*O(x^n) )^2 )^1; polcoeff(x^2*A, n+2)}

%o for(n=-2,60,print1(a(n),", "))

%o (PARI) {a(n) = my(A); A = sum(m=-2*n-4,n+2, ( x^m / (1 - x^(2*m)*(1+x)^(m+1)) +x^3*O(x^n) )^1 )^2; polcoeff(x^2*A, n+2)}

%o for(n=-2,60,print1(a(n),", "))

%K nonn

%O -2,5

%A _Paul D. Hanna_, Mar 09 2016