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A105634 Expansion of Sum_{k>0} Kronecker(k,7)*x^k*(1 + x^k)/(1 - x^k)^3. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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, Birkhauser, Boston, 1996, see page 107.

B. 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).

LINKS

Table of n, a(n) for n=1..50.

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.

EXAMPLE

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 + ...

PROG

(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)))}

CROSSREFS

Sequence in context: A169705 A138810 A331700 * A294124 A120036 A036381

Adjacent sequences:  A105631 A105632 A105633 * A105635 A105636 A105637

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Apr 16 2005, Mar 31 2008

STATUS

approved

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Last modified November 26 09:38 EST 2020. Contains 338639 sequences. (Running on oeis4.)