A206140 G.f.: Sum_{n>=0} x^n/Product_{k=1..n} (1 - Lucas(k)*x^k + (-1)^k*x^(2*k)).

1, 1, 2, 4, 10, 18, 44, 78, 178, 329, 699, 1299, 2724, 5013, 10198, 19076, 37786, 70448, 138322, 256649, 497884, 925309, 1770195, 3283971, 6249417, 11542969, 21785791, 40231877, 75388314, 138806128, 259010824, 475364152, 882643035, 1617443421, 2988026528

Offset: 0

Author

Paul D. Hanna, Feb 04 2012

Keywords

• nonn

Comments

Compare to the g.f. of partitions: Sum_{n>=0} x^n/Product_{k=1..n} (1-x^k).

As an analog to the identity: (1-x^n) = Product_{k=0..n-1} (1 - u^k*x), where u=exp(2*Pi*I/n), we have (1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Product_{k=0..n-1} (1 - u^k*x - (u^k*x)^2).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 18*x^5 + 44*x^6 + 78*x^7 +...
where
A(x) = 1 + x/(1-x-x^2) + x^2/((1-x-x^2)*(1-3*x^2+x^4)) + x^3/((1-x-x^2)*(1-3*x^2+x^4)*(1-4*x^3-x^6)) + x^4/((1-x-x^2)*(1-3*x^2+x^4)*(1-4*x^3-x^6)*(1-7*x^4+x^8)) + x^5/((1-x-x^2)*(1-3*x^2+x^4)*(1-4*x^3-x^6)*(1-7*x^4+x^8)*(1-11*x^5-x^10)) +...
The Lucas numbers begin: A000204 = [1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, ...].
		

Crossrefs

Cf. A000204, A206141.

Programs

PARI

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=0,n,x^m/prod(k=1,m,1-Lucas(k)*x^k+(-1)^k*x^(2*k)+x*O(x^n))),n)}
for(n=0,51,print1(a(n),", "))