A209455 a(n) = Pell(n)*A002652(n) for n>=1, with a(0)=1, where A002652 lists the coefficients in theta series of Kleinian lattice Z[(-1+sqrt(-7))/2].
1, 2, 8, 0, 72, 0, 0, 338, 3264, 1970, 0, 22964, 0, 0, 323128, 0, 4708320, 0, 10976840, 0, 0, 0, 745778864, 900234724, 0, 2623476242, 0, 0, 110745336312, 178241928596, 0, 0, 7524162792576, 0, 0, 0, 127800022137480, 205691031143924, 0, 0, 0, 0, 0, 40725785296405556
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 2*x + 8*x^2 + 72*x^4 + 338*x^7 + 3264*x^8 + 1970*x^9 +... where A(x) = 1 + 1*2*x + 2*4*x^2 + 12*6*x^4 + 169*2*x^7 + 408*8*x^8 + 985*2*x^9 + 5741*4*x^11 + 80782*4*x^14 + 470832*10*x^16 +...+ Pell(n)*A002652(n)*x^n +... The g.f. is also given by the identity: A(x) = 1 + 2*( 1*x/(1-2*x-x^2) + 2*x^2/(1-6*x^2+x^4) - 5*x^3/(1-14*x^3-x^6) + 12*x^4/(1-34*x^4+x^8) - 29*x^5/(1-82*x^5-x^10) - 70*x^6/(1-198*x^6+x^12) + 0*169*13*x^7/(1+478*x^7-x^14) +...). The values of the symbol Kronecker(n,7) repeat [1,1,-1,1,-1,-1,0, ...].
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
terms = 44; s = 1 + 2 Sum[x^n*Fibonacci[n, 2]*KroneckerSymbol[n, 7]/(1 + (-1)^n*x^(2*n) - x^n*(Fibonacci[n - 1, 2] + Fibonacci[n + 1, 2])), {n, 1, terms}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Jul 05 2017 *) A002652[n_]:= If[n < 1, Boole[n == 0], 2*Sum[KroneckerSymbol[-7, d], {d, Divisors[n]}]]; Join[{1}, Table[Fibonacci[n, 2]*A002652[n], {n,1,50}]] (* G. C. Greubel, Jan 03 2017 *) -
PARI
{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)} {A002203(n)=Pell(n-1)+Pell(n+1)} {a(n)=polcoeff(1 + 2*sum(m=1,n,Pell(m)*kronecker(m,7)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)} for(n=0,60,print1(a(n),", "))
Formula
G.f.: 1 + 2*Sum_{n>=1} Pell(n)*Kronecker(n,7)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).
G.f.: 1 + 2*Sum_{n>=1} F(n,2)*Kronecker(n,7)*x^n/(1 + (-1)^n*x^(2*n)-x^n* (F(n-1,2)+F(n+1,2))), where F is the Fibonacci polynomial. - Jean-François Alcover, Jul 05 2017
Comments