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%I #25 Nov 16 2023 07:44:13
%S 1,5,8,21,24,40,49,85,73,120,122,168,168,245,192,341,288,365,360,504,
%T 392,610,530,680,601,840,656,1029,842,960,960,1365,976,1440,1176,1533,
%U 1370,1800,1344,2040,1680,1960,1850,2562,1752,2650,2208,2728,2401,3005
%N Expansion of Sum_{k>0} Kronecker(k,7)*x^k*(1 + x^k)/(1 - x^k)^3.
%D 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.
%D 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).
%F 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).
%F G.f.: Sum_{k>0} Kronecker(k, 7)*x^k*(1+x^k)/(1-x^k)^3.
%F a(n) = A002656(n) + 8*A053724(n-2).
%F a(7n) = 49a(n).
%F 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.
%F 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
%e 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 + ...
%t 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 *)
%o (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)))))) }
%o (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))}
%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d^2 * kronecker(-7, n / d)))}
%Y Cf. A002656, A053724, A138809, A327135.
%K nonn,easy,mult
%O 1,2
%A _Michael Somos_, Apr 16 2005, Mar 31 2008
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