A204274 - OEIS

%I #15 Mar 25 2019 11:15:27

%S 1,0,0,12,0,0,0,0,985,0,0,0,0,0,0,470832,0,0,0,0,0,0,0,0,1311738121,0,

%T 0,0,0,0,0,0,0,0,0,21300003689580,0,0,0,0,0,0,0,0,0,0,0,0,

%U 2015874949414289041,0,0,0,0,0,0,0,0,0

%N G.f.: Sum_{n>=1} Pell(n^2)*x^(n^2).

%C Compare g.f. to the Lambert series identity: Sum_{n>=1} lambda(n)*x^n/(1-x^n) = Sum_{n>=1} x^(n^2); Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity).

%H Robert Israel, <a href="/A204274/b204274.txt">Table of n, a(n) for n = 1..2500</a>

%F G.f.: Sum_{n>=1} lambda(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where lambda(n) = A008836(n), Pell(n) = A000129(n) and A002203 is the companion Pell numbers.

%e G.f.: A(x) = x + 12*x^4 + 985*x^9 + 470832*x^16 + 1311738121*x^25 +...

%e where A(x) = x/(1-2*x-x^2) + (-1)*2*x^2/(1-6*x^2+x^4) + (-1)*5*x^3/(1-14*x^3-x^6) + (+1)*12*x^4/(1-34*x^4+x^8) + (-1)*29*x^5/(1-82*x^5-x^10) + (+1)*70*x^6/(1-198*x^6+x^12) +...+ lambda(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...

%p pell:= gfun:-rectoproc({a(0)=0,a(1)=1,a(n)=2*a(n-1)+a(n-2)},a(n),remember):

%p seq(`if`(issqr(n),pell(n),0), n=1..100); # _Robert Israel_, Nov 24 2015

%t CoefficientList[Sum[Fibonacci[n^2, 2] x^n^2/x, {n, 1, 8}], x] (* _Jean-François Alcover_, Mar 25 2019 *)

%o (PARI) /* Subroutines used in PARI programs below: */

%o {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}

%o {A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}

%o (PARI) {a(n)=issquare(n)*Pell(n)}

%o (PARI) {lambda(n)=local(F=factor(n));(-1)^sum(i=1,matsize(F)[1],F[i,2])}

%o {a(n)=polcoeff(sum(m=1,n,lambda(m)*Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}

%Y Cf. A204327, A204060, A204270, A204271, A204272, A204273, A204275, A008836 (lambda), A002203, A000045.

%K nonn

%O 1,4

%A _Paul D. Hanna_, Jan 14 2012