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A225524 G.f.: exp( Sum_{n>=1} (sigma(2*n) - sigma(n))*Lucas(n)*x^n/n ), where Lucas(n) = A000204(n) and sigma(n) = A000203(n) is the sum of divisors of n. 2

%I #10 May 09 2013 21:03:20

%S 1,2,8,24,66,184,488,1248,3136,7776,18780,44880,105896,246124,567008,

%T 1293840,2920626,6545352,14555388,32115120,70421792,153451488,

%U 332314512,715843200,1534016392,3270661294,6941489040,14667591672,30859685088,64670865304,135011595856,280813639680

%N G.f.: exp( Sum_{n>=1} (sigma(2*n) - sigma(n))*Lucas(n)*x^n/n ), where Lucas(n) = A000204(n) and sigma(n) = A000203(n) is the sum of divisors of n.

%C Compare g.f. to theta_4(x) = exp( Sum_{n>=1} -(sigma(2*n)-sigma(n))*x^n/n ), where Jacobi theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n*x^(n^2).

%F G.f.: Product_{n>=1} (1 + Lucas(n)*x^n + (-x^2)^n) / (1 - Lucas(n)*x^n + (-x^2)^n).

%F G.f.: 1/Product_{n>=1} (1 - Lucas(2*n-1)*x^(2*n-1) - x^(4*n-2))^2 * (1 - Lucas(2*n)*x^(2*n) + x^(4*n)).

%F Logarithmic derivative equals A225525.

%e G.f.: A(x) = 1 + 2*x + 8*x^2 + 24*x^3 + 66*x^4 + 184*x^5 + 488*x^6 + 1248*x^7 +...

%e The g.f. equals the product:

%e A(x) = (1+x-x^2)/(1-x-x^2) * (1+3*x^2+x^4)/(1-3*x^2+x^4) * (1+4*x^3-x^6)/(1-4*x^3-x^6) * (1+7*x^4+x^8)/(1-7*x^4+x^8) * (1+11*x^5-x^10)/(1-11*x^5-x^10) *...* (1 + Lucas(n)*x^n + (-x^2)^n)/(1 - Lucas(n)*x^n + (-x^2)^n) *...

%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

%o {a(n)=polcoeff(prod(m=1, n, 1 + Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))/prod(m=1, n, 1 - Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n)), n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

%o {a(n)=polcoeff(1/prod(m=1, n\2+1, (1 - Lucas(2*m-1)*x^(2*m-1) - x^(4*m-2))^2*(1 - Lucas(2*m)*x^(2*m) + x^(4*m) +x*O(x^n))), n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

%o {a(n)=polcoeff(exp(sum(k=1, n, (sigma(2*k)-sigma(k))*Lucas(k)*x^k/k)+x*O(x^n)), n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A203850, A225525.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 09 2013

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Last modified March 29 06:15 EDT 2024. Contains 371265 sequences. (Running on oeis4.)