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A156234 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*A000204(n)*x^n/n ).

%I A156234 #16 Dec 29 2015 18:08:01
%S A156234 1,1,5,10,30,63,170,355,880,1875,4349,9189,20810,43355,95140,198247,
%T A156234 424527,875965,1849535,3781820,7873167,16005196,32883560,66390850,
%U A156234 135198990,271051271,546931398,1090751095,2183512495,4329540830
%N A156234 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*A000204(n)*x^n/n ).
%C A156234 Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ),
%C A156234 and to the g.f. of Fibonacci numbers: exp( Sum_{n>=1} A000204(n)*x^n/n ) where A000204 is the Lucas numbers.
%H A156234 Robert Israel, <a href="/A156234/b156234.txt">Table of n, a(n) for n = 0..3000</a>
%F A156234 a(n) = (1/n)*Sum_{k=1..n} sigma(n)*A000204(k)*a(n-k) for n>0, with a(0) = 1.
%F A156234 G.f.: Product_{n>=1} 1/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000204(n).
%F A156234 Logarithmic derivative yields A225528.
%e A156234 G.f.: A(x) = 1 + x + 5*x^2 + 10*x^3 + 30*x^4 + 63*x^5 + 170*x^6 + 355*x^7 + ...
%e A156234 log(A(x)) = x + 3*3*x^2/2 + 4*4*x^3/3 + 7*7*x^4/4 + 6*11*x^5/5 + 12*18*x^6/6 + ...
%e A156234 Also, the g.f. equals the product:
%e A156234 A(x) = 1/((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) * (1-18*x^6+x^12) * ... * (1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) * ...).
%p A156234 N:= 100: # to get a(0) to a(N)
%p A156234 G:= exp(add(numtheory:-sigma(n)*lucas(n)*x^n/n,n=1..N)):
%p A156234 S:= series(G,x,N+1):
%p A156234 seq(coeff(S,x,i),i=0..N); # _Robert Israel_, Dec 23 2015
%o A156234 (PARI) {a(n)=polcoeff(exp(sum(k=1,n,sigma(k)*(fibonacci(k-1)+fibonacci(k+1))*x^k/k)+x*O(x^n)),n)}
%o A156234 for(n=0,40,print1(a(n),", "))
%o A156234 (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A156234 {a(n)=polcoeff(prod(m=1,n,1/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
%o A156234 for(n=0,40,print1(a(n),", "))
%Y A156234 Cf. A225528, A000203 (sigma), A000204 (Lucas), A000041 (partitions), A000045.
%K A156234 nonn
%O A156234 0,3
%A A156234 _Paul D. Hanna_, Feb 06 2009