A204274 - OEIS

A204274

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

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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21300003689580, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2015874949414289041, 0, 0, 0, 0, 0, 0, 0, 0, 0

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OFFSET

1,4

COMMENTS

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).

LINKS

Robert Israel, Table of n, a(n) for n = 1..2500

FORMULA

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.

EXAMPLE

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

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)) +...

MAPLE

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

seq(`if`(issqr(n), pell(n), 0), n=1..100); # Robert Israel, Nov 24 2015

MATHEMATICA

CoefficientList[Sum[Fibonacci[n^2, 2] x^n^2/x, {n, 1, 8}], x] (* Jean-François Alcover, Mar 25 2019 *)

PROG

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

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

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

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

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

{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)}