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A035170 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -20. 9
1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 0, 2, 0, 2, 2, 1, 0, 3, 0, 1, 4, 0, 2, 2, 1, 0, 4, 2, 2, 2, 0, 1, 0, 0, 2, 3, 0, 0, 0, 1, 2, 4, 2, 0, 3, 2, 2, 2, 3, 1, 0, 0, 0, 4, 0, 2, 0, 2, 0, 2, 2, 0, 6, 1, 0, 0, 2, 0, 4, 2, 0, 3, 0, 0, 2, 0, 0, 0, 0, 1, 5, 2, 2, 4, 0, 2, 4, 0, 2, 3, 0, 2, 0, 2, 0, 2, 0, 3, 0, 1, 2, 0, 2, 0, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

REFERENCES

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 253.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Multiplicative with a(2^e) = a(5^e) = 1, a(p^e) = e+1 if p == 1, 3, 7, 9 (mod 20), a(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20). - Michael Somos, Sep 10 2005

G.f.: Sum_{k>0} x^k * (1 + x^(2*k)) * (1 + x^(6*k)) / (1 + x^(10*k)). - Michael Somos, Sep 10 2005

a(2*n) = a(5*n) = a(n), a(20*n + 11) = a(20*n + 13) = a(20*n + 17) = a(20*n + 19) = 0.

Moebius transform is period 20 sequence [ 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, ...]. - Michael Somos, Oct 21 2006

Expansion of -1 + (phi(q) * phi(q^5) + phi(q^2) * phi(q^10) + 4 * q^3 * psi(q^4)* psi(q^20)) / 2 in powers of q where phi(), psi() are Ramanujan theta functions.

2*a(n) = A028586(n) + A033718(n) if n>0. - Michael Somos, Oct 21 2006

a(n) = A124233(n) unless n=0. a(n) = |A111949(n)|. a(2*n + 1) = A129390(n). a(4*n + 3) = 2 * A033764(n).

EXAMPLE

q + q^2 + 2*q^3 + q^4 + q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + q^10 + ...

MATHEMATICA

QP = QPochhammer; s = (1/q) * (QP[q^2]*QP[q^4]*QP[q^5]*(QP[q^10] / (QP[q]* QP[q^20]))-1) + O[q]^105; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Dec 04 2015 *)

a[n_] := If[n < 0, 0, DivisorSum[ n, KroneckerSymbol[-20, #] &]]; Table[a[n], {n, 1, 100}] (* G. C. Greubel, Dec 12 2017 *)

PROG

(PARI) direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X))

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -20, d)))} /* Michael Somos, Sep 10 2005 */

(PARI) {a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( -20, p) * X) )[n])} /* Michael Somos, Sep 10 2005 */

(PARI) {a(n) = if( n<1, 0, qfrep([1, 0; 0, 5], n)[n] + qfrep([2, 1; 1, 3], n)[n])} /* Michael Somos, Oct 21 2006 */

CROSSREFS

Cf. A028586, A033718, A033764, A111949, A124233, A129390.

Sequence in context: A029313 A144001 A124233 * A111949 A143323 A086598

Adjacent sequences:  A035167 A035168 A035169 * A035171 A035172 A035173

KEYWORD

nonn,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 23 22:45 EDT 2019. Contains 326254 sequences. (Running on oeis4.)