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A205884
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a(n) = Pell(n)*A109064(n) for n >= 1 with a(0)=1.
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3
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1, -5, 10, 50, -180, -145, -700, 5070, 10200, -34475, 11890, -344460, 415800, 2007660, -2423460, 1950250, -25895760, 90935120, 96047350, -662510900, -239916420, -2316837900, 5593341480, 24756454910, -27166986000, -6558690605, -190008957720, 764537004500
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OFFSET
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0,2
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COMMENTS
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Compare to g.f. of A109064, which is the Lambert series identity:
1 - 5*Sum_{n>=1} L(n,5)*n*x^n/(1-x^n) = eta(x)^5/eta(x^5).
Here L(n,5) is the Legendre symbol given by A080891(n).
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LINKS
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FORMULA
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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.
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EXAMPLE
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G.f.: A(x) = 1 - 5*x + 10*x^2 + 50*x^3 - 180*x^4 - 145*x^5 - 700*x^6 + ...
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 + ...
The g.f. is illustrated by:
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) + ...
The values of the Legendre symbol L(n,5) repeat: [1,-1,-1,1,0, ...].
The companion Pell numbers (A002203) begin: [2,6,14,34,82,198,478,1154,2786,6726,16238,39202,94642,...].
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MATHEMATICA
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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 *)
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PROG
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(PARI) {A109064(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^5/eta(x^5+A), n))}
{a(n)=if(n==0, 1, Pell(n)*A109064(n))}
(PARI) {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)}
{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)}
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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