A105634 Expansion of Sum_{k>0} Kronecker(k,7)*x^k*(1 + x^k)/(1 - x^k)^3.

1, 5, 8, 21, 24, 40, 49, 85, 73, 120, 122, 168, 168, 245, 192, 341, 288, 365, 360, 504, 392, 610, 530, 680, 601, 840, 656, 1029, 842, 960, 960, 1365, 976, 1440, 1176, 1533, 1370, 1800, 1344, 2040, 1680, 1960, 1850, 2562, 1752, 2650, 2208, 2728, 2401, 3005

Offset: 1

Author

Michael Somos, Apr 16 2005, Mar 31 2008

Keywords

• nonn
• easy
• mult

Examples

			q + 5*q^2 + 8*q^3 + 21*q^4 + 24*q^5 + 40*q^6 + 49*q^7 + 85*q^8 + 73*q^9 + ...
		

References

• A. Balog, H. Darmon and K. Ono, Congruence for Fourier coefficients of half-integral weight modular forms and special values of L-functions, pp. 105-128 of Analytic number theory, Vol. 1, Birkhäuser, Boston, 1996, see page 107.
• Bruce Berndt, Commentary on Ramanujan's Papers, pp. 357-426 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea, 2000. See page 372 (4).

Crossrefs

Cf. A002656, A053724, A138809, A327135.

Programs

Mathematica

f[p_, e_] := If[MemberQ[{1, 2, 4}, Mod[p, 7]], (p^(2*e+2)-1)/(p^2-1), (p^(2*e+2)+(-1)^e)/(p^2+1)]; f[7, e_] := 7^(2*e); a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 04 2023 *)

PARI

{a(n)=local(A,p,e); if(n<2, n==1, A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==7, p^(2*e), if(kronecker(p,7)==1, (p^(2*e+2)-1)/(p^2-1), (p^(2*e+2)+(-1)^e)/(p^2+1)))))) }

PARI

{a(n)=local(A,B); if(n<1, 0, n--; A=x*O(x^n); polcoeff( if(B=eta(x^7+A), A=eta(x+A); (A*B)^3+8*x*B^7/A), n))}

PARI

{a(n) = if( n<1, 0, sumdiv(n, d, d^2 * kronecker(-7, n / d)))}

Formula

Multiplicative with a(p^e) = p^(2e) if p = 7; (p^(2e+2)-1)/(p^2-1) if p == 1, 2, 4 (mod 7); (p^(2e+2)+(-1)^e)/(p^2+1) if p == 3, 5, 6 (mod 7).

G.f.: Sum_{k>0} Kronecker(k, 7)*x^k*(1+x^k)/(1-x^k)^3.

a(n) = A002656(n) + 8*A053724(n-2).

a(7n) = 49a(n).

G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 7^(-1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is g.f. for A138809.

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 32*Pi^3/(343*sqrt(7)) = 1.093343069... (A327135). - Amiram Eldar, Nov 16 2023