A205884 - OEIS

%I #26 Dec 14 2017 03:46:25

%S 1,-5,10,50,-180,-145,-700,5070,10200,-34475,11890,-344460,415800,

%T 2007660,-2423460,1950250,-25895760,90935120,96047350,-662510900,

%U -239916420,-2316837900,5593341480,24756454910,-27166986000,-6558690605,-190008957720,764537004500

%N a(n) = Pell(n)*A109064(n) for n >= 1 with a(0)=1.

%C Compare to g.f. of A109064, which is the Lambert series identity:

%C 1 - 5*Sum_{n>=1} L(n,5)*n*x^n/(1-x^n) = eta(x)^5/eta(x^5).

%C Here L(n,5) is the Legendre symbol given by A080891(n).

%H G. C. Greubel, <a href="/A205884/b205884.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1 - 5*Sum_{n>=1} Pell(n)*L(n,5)*n*x^n / (1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where L(n,5) is the Legendre symbol, Pell(n) = A000129(n), and A002203 is the companion Pell numbers.

%e G.f.: A(x) = 1 - 5*x + 10*x^2 + 50*x^3 - 180*x^4 - 145*x^5 - 700*x^6 + ...

%e where A(x) = 1 - 1*5*x + 2*5*x^2 + 5*10*x^3 - 12*15*x^4 - 29*5*x^5 - 70*10*x^6 + 169*30*x^7 + 408*25*x^8 + ... + Pell(n)*A109064(n)*x^n + ...

%e The g.f. is illustrated by:

%e A(x) = 1 - 5*(+1)*1*1*x/(1-2*x-x^2) - 5*(-1)*2*2*x^2/(1-6*x^2+x^4) - 5*(-1)*3*5*x^3/(1-14*x^3-x^6) - 5*(+1)*4*12*x^4/(1-34*x^4+x^8) - 5*(0)*5*29*x^5/(1-82*x^5-x^10) - 5*(+1)*6*70*x^6/(1-198*x^6+x^12) + ...

%e The values of the Legendre symbol L(n,5) repeat: [1,-1,-1,1,0, ...].

%e The companion Pell numbers (A002203) begin: [2,6,14,34,82,198,478,1154,2786,6726,16238,39202,94642,...].

%t pell[n_] := ((1+Sqrt[2])^n - (1-Sqrt[2])^n)/(2*Sqrt[2]) // Simplify; (* b = A109064 *); b[0] = 1; b[n_] := b[n] = Sum[DivisorSum[j, #*If[Divisible[#, 5], -4, -5] &]*b[n - j], {j, 1, n}]/n; a[0] = 1; a[n_] := pell[n]*b[n]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Apr 24 2017 *)

%o (PARI) {A109064(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^5/eta(x^5+A), n))}

%o {a(n)=if(n==0,1,Pell(n)*A109064(n))}

%o (PARI) {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 {a(n)=polcoeff(1-5*sum(m=1, n, kronecker(m, 5)*m*Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}

%Y Cf. A109064, A080891, A000129 (Pell), A002203 (companion Pell), A205882 (variant), A204270.

%K sign

%O 0,2

%A _Paul D. Hanna_, Feb 01 2012