A205968
a(n) = Fibonacci(n)*A008655(n) for n >= 1, with a(0)=1, where A008655 lists the coefficients in (theta_3(x)*theta_3(3*x) + theta_2(x)*theta_2(3*x))^4.
Original entry on oeis.org
1, 24, 216, 1776, 5256, 15120, 63936, 107328, 294840, 823344, 1496880, 2845152, 9334656, 12291216, 28012608, 68251680, 110883528, 188343792, 563167296, 688359840, 1493387280, 3343696512, 5095667232, 8368761024, 24087248640, 28361250600, 57633511824, 128471514096
Offset: 0
G.f.: A(x) = 1 + 24*x + 216*x^2 + 1776*x^3 + 5256*x^4 + 15120*x^5 + ...
where A(x) = 1 + 1*24*x + 1*216*x^2 + 2*888*x^3 + 3*1752*x^4 + 5*3024*x^5 + ... + Fibonacci(n)*A008655(n)*x^n + ...
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A008655 := CoefficientList[Series[((EllipticTheta[3, 0, q]^3 + EllipticTheta[3, Pi/3, q]^3 + EllipticTheta[3, 2 Pi/3, q]^3)^4/(3* EllipticTheta[3, 0, q^3])^4), {q, 0, 250}], q]; b := Table[A008655[[n]], {n, 1, 120}][[1 ;; ;; 2]]; Join[{1}, Table[Fibonacci[n]*b[[n + 1]], {n, 1, 50}]] (* G. C. Greubel, Jul 16 2018 *)
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{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + sum(m=1,n, 24*fibonacci(m)*m^3*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n)) + 8*fibonacci(3*m)*(3*m)^3*x^(3*m)/(1-Lucas(3*m)*x^(3*m)+(-1)^m*x^(6*m) +x*O(x^n)) ),n)}
for(n=0,40,print1(a(n),", "))
A209447
a(n) = Pell(n)*A008653(n) for n>=1, with a(0)=1, where A008653 is the theta series of direct sum of 2 copies of hexagonal lattice.
Original entry on oeis.org
1, 12, 72, 60, 1008, 2088, 2520, 16224, 73440, 11820, 513648, 826704, 1164240, 5621448, 23265216, 14041800, 175149504, 245524824, 98791560, 1590026160, 8061191712, 3706940640, 40272058656, 64816900128, 97801149600, 487966581012, 1596075244848, 91744440540
Offset: 0
G.f.: A(x) = 1 + 12*x + 72*x^2 + 60*x^3 + 1008*x^4 + 2088*x^5 + 2520*x^6 +...
where A(x) = 1 + 1*12*x + 2*36*x^2 + 5*12*x^3 + 12*84*x^4 + 29*72*x^5 + 70*36*x^6 +...+ Pell(n)*A008653(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 12*( 1*1*x/(1-2*x-x^2) + 2*2*x^2/(1-6*x^2+x^4) + 12*4*x^4/(1-34*x^4+x^8) + 29*5*x^5/(1-82*x^5-x^10) + 169*7*x^7/(1-478*x^7-x^14) + 408*8*x^8/(1-1154*x^8-x^16) +...).
The values of the Dirichlet character Chi(n,3) repeat [1,1,0, ...].
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A008653[n_]:= If[n < 1, Boole[n == 0], 12*Sum[If[Mod[d, 3] > 0, d, 0], {d, Divisors@n}]]; Join[{1}, Table[Fibonacci[n, 2]*A008653[n], {n, 1, 1000}]] (* G. C. Greubel, Jan 02 2017 *)
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{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 + 12*sum(m=1,n,Pell(m)*kronecker(m,3)^2*m*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,50,print1(a(n),", "))
A209449
a(n) = Pell(n)*A113973(n) for n>=1, with a(0)=1, where A113973 lists the coefficients in phi(x^3)^3/phi(x) and phi() is a Ramanujan theta function.
Original entry on oeis.org
1, -2, 8, -10, 24, 0, 280, -676, 1632, -1970, 0, 0, 27720, -133844, 646256, 0, 941664, 0, 10976840, -26500436, 0, -154455860, 0, 0, 2173358880, -2623476242, 25334527696, -15290740090, 73830224208, 0, 0, -1038870091396, 2508054264192, 0, 0, 0, 42600007379160
Offset: 0
G.f.: A(x) = 1 - 2*x + 8*x^2 - 10*x^3 + 24*x^4 + 280*x^6 - 676*x^7 +...
where A(x) = 1 - 1*2*x + 2*4*x^2 - 5*2*x^3 + 12*2*x^4 + 70*4*x^6 - 169*4*x^7 + 408*4*x^8 +...+ Pell(n)*A113973(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) + 12*x^4/(1-34*x^4+x^8) - 29*x^5/(1+82*x^5-x^10) + 169*x^7/(1+478*x^7-x^14) - 408*x^8/(1-1154*x^8+x^16) +...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].
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A113973:= CoefficientList[Series[EllipticTheta[3, 0, q^3]^3 /EllipticTheta[3, 0, q], {q, 0, 60}], q]; Table[If[n == 0, 1, Fibonacci[n, 2]*A113973[[n + 1]]], {n, 0, 50}] (* G. C. Greubel, Dec 03 2017 *)
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{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,3)*x^m/(1-A002203(m)*(-x)^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,60,print1(a(n),", "))
Showing 1-3 of 3 results.
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