A205974 a(n) = Fibonacci(n)*A033719(n) for n>=1, with a(0)=1, where A033719 lists the coefficients in theta_3(q)*theta_3(q^7).
1, 2, 0, 0, 6, 0, 0, 26, 84, 68, 0, 356, 0, 0, 0, 0, 5922, 0, 0, 0, 0, 0, 0, 114628, 0, 150050, 0, 0, 635622, 2056916, 0, 0, 17426472, 0, 0, 0, 29860704, 96631268, 0, 0, 0, 0, 0, 1733977748, 2805634932, 0, 0, 0, 0, 15557484098, 0, 0, 0, 213265164692, 0, 0
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 2*x + 6*x^4 + 26*x^7 + 84*x^8 + 68*x^9 + 356*x^11 +... where A(x) = 1 + 1*2*x + 3*2*x^4 + 13*2*x^7 + 21*4*x^8 + 34*2*x^9 + 89*4*x^11 + 987*6*x^16 + 28657*4*x^23 +...+ Fibonacci(n)*A033719(n)*x^n +... The g.f. is also given by the identity: A(x) = 1 + 2*( 1*x/(1+x-x^2) + 1*x^2/(1-3*x^2+x^4) - 2*x^3/(1+4*x^3-x^6) + 3*x^4/(1-7*x^4+x^8) - 5*x^5/(1+11*x^5-x^10) - 8*x^6/(1-18*x^6+x^12) + 0*13*x^7/(1+29*x^7-x^14) +...). The values of the symbol Kronecker(n,7) repeat [1,1,-1,1,-1,-1,0, ...].
Programs
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PARI
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)} {a(n)=polcoeff(1 + 2*sum(m=1,n,fibonacci(m)*kronecker(m,7)*x^m/(1-Lucas(m)*(-x)^m+(-1)^m*x^(2*m) +x*O(x^n))),n)} for(n=0,40,print1(a(n),", "))
Formula
G.f.: 1 + 2*Sum_{n>=1} Fibonacci(n)*Kronecker(n,7)*x^n/(1 - Lucas(n)*(-x)^n + (-1)^n*x^(2*n)).
Comments