|
%I #11 Dec 04 2017 03:01:36
%S 1,-2,8,-10,24,0,280,-676,1632,-1970,0,0,27720,-133844,646256,0,
%T 941664,0,10976840,-26500436,0,-154455860,0,0,2173358880,-2623476242,
%U 25334527696,-15290740090,73830224208,0,0,-1038870091396,2508054264192,0,0,0,42600007379160
%N 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.
%C Compare g.f. to the Lambert series of A113973:
%C 1 - 2*Sum_{n>=1} Kronecker(n,3)*x^n/(1 - (-x)^n).
%H G. C. Greubel, <a href="/A209449/b209449.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: 1 - 2*Sum_{n>=1} Pell(n)*Kronecker(n,3)*x^n/(1 - A002203(n)*(-x)^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).
%e G.f.: A(x) = 1 - 2*x + 8*x^2 - 10*x^3 + 24*x^4 + 280*x^6 - 676*x^7 +...
%e 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 +...
%e The g.f. is also given by the identity:
%e 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) +...).
%e The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].
%t 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 *)
%o (PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
%o {A002203(n)=Pell(n-1)+Pell(n+1)}
%o {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)}
%o for(n=0,60,print1(a(n),", "))
%Y Cf. A113973, A205969, A209446, A209448, A209450, A204270, A000129 (Pell), A002203.
%K sign
%O 0,2
%A _Paul D. Hanna_, Mar 10 2012
|
