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

%I A206140 #15 Feb 17 2025 08:14:47
%S A206140 1,1,2,4,10,18,44,78,178,329,699,1299,2724,5013,10198,19076,37786,
%T A206140 70448,138322,256649,497884,925309,1770195,3283971,6249417,11542969,
%U A206140 21785791,40231877,75388314,138806128,259010824,475364152,882643035,1617443421,2988026528
%N A206140 G.f.: Sum_{n>=0} x^n/Product_{k=1..n} (1 - Lucas(k)*x^k + (-1)^k*x^(2*k)).
%C A206140 Compare to the g.f. of partitions: Sum_{n>=0} x^n/Product_{k=1..n} (1-x^k).
%C A206140 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).
%e A206140 G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 18*x^5 + 44*x^6 + 78*x^7 +...
%e A206140 where
%e A206140 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)) +...
%e A206140 The Lucas numbers begin: A000204 = [1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, ...].
%o A206140 (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A206140 {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)}
%o A206140 for(n=0,51,print1(a(n),", "))
%Y A206140 Cf. A000204, A206141.
%K A206140 nonn
%O A206140 0,3
%A A206140 _Paul D. Hanna_, Feb 04 2012