A292498 - OEIS

%I #19 Oct 10 2017 07:14:57

%S 1,1,2,3,3,8,10,21,18,67,54,144,196,377,470,1077,1321,2584,3905,6765,

%T 10014,18173,27084,46368,73001,121448,191530,319827,505038,832040,

%U 1335766,2178309,3497250,5710862,9183554,14931192,24093521,39088169,63117470,102363639,165353391,267914296,433177813,701408733,1134249308,1836422951,2970148632

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

%C Compare g.f. to: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n  =  0.

%H Paul D. Hanna, <a href="/A292498/b292498.txt">Table of n, a(n) for n = 0..1030</a>

%F G.f. P(x) + Q(x) where

%F P(x) = Sum_{n>=0} Fibonacci(n+1) * x^n * (1-x^n)^n,

%F Q(x) = Sum_{n>=1} Fibonacci(n-1) * x^(n^2-n) / (1-x^n)^n.

%F a(n) ~ phi^(n+1) / sqrt(5), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Oct 10 2017

%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 3*x^4 + 8*x^5 + 10*x^6 + 21*x^7 + 18*x^8 + 67*x^9 + 54*x^10 + 144*x^11 + 196*x^12 + 377*x^13 + 470*x^14 + 1077*x^15 + 1321*x^16 + 2584*x^17 + 3905*x^18 + 6765*x^19 + 10014*x^20 +...

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

%e RELATED SERIES.

%e G.f. A(x) = P(x) + Q(x) where

%e P(x) = 1 + 1*x*(1-x) + 2*x^2*(1-x^2)^2 + 3*x^3*(1-x^3)^3 + 5*x^4*(1-x^4)^4 + 8*x^5*(1-x^5)^5 + 13*x^6*(1-x^6)^6 + 21*x^7*(1-x^7)^7 + 34*x^8*(1-x^8)^8 + 55*x^9*(1-x^9)^9 + 89*x^10*(1-x^10)^10 +...+ Fibonacci(n+1)*x^n*(1-x^n)^n +...

%e Q(x) = 0*x^0/(1-x) + 1*x^2/(1-x^2)^2 + 1*x^6/(1-x^3)^3 + 2*x^12/(1-x^4)^4 + 3*x^20/(1-x^5)^5 + 5*x^30/(1-x^6)^6 + 8*x^42/(1-x^7)^7 + 13*x^56/(1-x^8)^8 + 21*x^72/(1-x^9)^9 +...+ Fibonacci(n-1)*x^(n^2-n)/(1-x^n)^n +...

%e Explicitly,

%e P(x) = 1 + x + x^2 + 3*x^3 + x^4 + 8*x^5 + 6*x^6 + 21*x^7 + 14*x^8 + 64*x^9 + 49*x^10 + 144*x^11 + 182*x^12 + 377*x^13 + 463*x^14 + 1067*x^15 + 1305*x^16 + 2584*x^17 + 3881*x^18 + 6765*x^19 + 9981*x^20 + 18152*x^21 +...

%e Q(x) = x^2 + 2*x^4 + 4*x^6 + 4*x^8 + 3*x^9 + 5*x^10 + 14*x^12 + 7*x^14 + 10*x^15 + 16*x^16 + 24*x^18 + 33*x^20 + 21*x^21 + 11*x^22 + 80*x^24 + 15*x^25 +...

%e The reciprocal of the g.f. begins:

%e 1/A(x) = 1 - x - x^2 + 2*x^4 - 4*x^5 + x^6 - 2*x^7 + 19*x^8 - 50*x^9 + 58*x^10 - 42*x^11 + 75*x^12 - 267*x^13 + 566*x^14 - 827*x^15 + 1225*x^16 - 2431*x^17 + 4972*x^18 - 8438*x^19 + 13089*x^20 - 22333*x^21 + 43831*x^22 +...

%o (PARI) {a(n) = my(A,P,Q,Ox=x*O(x^n));

%o P = sum(m=0,n, fibonacci(m+1) * x^m * (1-x^m +Ox)^m);

%o Q = sum(m=1,sqrtint(2*n+9), fibonacci(m-1) * x^(m^2-m) / (1-x^m +Ox)^m);

%o A = P + Q; polcoeff(A,n)}

%o for(n=0,60,print1(a(n),", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 09 2017