login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A097195 Expansion of s(12)^3*s(18)^2/(s(6)^2*s(36)), where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. Then replace q^6 with q. 34
1, 2, 2, 2, 1, 2, 2, 2, 3, 0, 2, 2, 2, 2, 0, 4, 2, 2, 2, 0, 1, 2, 4, 2, 0, 2, 2, 2, 3, 2, 2, 0, 2, 2, 0, 2, 4, 2, 2, 0, 2, 4, 0, 4, 0, 2, 2, 2, 1, 0, 4, 2, 2, 0, 2, 2, 2, 4, 2, 0, 3, 2, 2, 2, 0, 0, 2, 4, 2, 0, 2, 4, 2, 2, 0, 0, 2, 2, 4, 2, 4, 2, 0, 2, 0, 4, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 80, Eq. (32.38).

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000

FORMULA

Expansion of q^(-1/6) * eta(q^2)^3 * eta(q^3)^2 / (eta(q)^2 * eta(q^6)) in powers of q. - Michael Somos, Mar 05 2016

Euler transform of period 6 sequence [ 2, -1, 0, -1, 2, -2, ...]. - Michael Somos, Mar 05 2016

Fine gives an explicit formula for a(n) in terms of the divisors of n.

a(n) = b(6*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).

G.f.: Sum_{k} x^k / (1 - x^(6*k + 1)). - Michael Somos, Nov 03 2005

G.f.: Sum_{k>=0} a(k) * x^(6*k + 1) = Sum_{k>0} x^(2*k-1) * (1 - x^(4*k - 2)) * (1 - x^(8*k - 4)) * (1 - x^(20*k - 10)) / (1 - x^(36*k - 18)). - Michael Somos, Nov 03 2005

6 * a(n) = A004016(6*n + 1). - Michael Somos, Mar 05 2016

EXAMPLE

G.f. = 1 + 2*x + 2*x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 3*x^8 + ...

G.f. = q + 2*q^7 + 2*q^13 + 2*q^19 + q^25 + 2*q^31 + 2*q^37 + 2*q^43 + ...

MATHEMATICA

a[n_] := DivisorSum[6n+1, KroneckerSymbol[-3, #]&]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Nov 23 2015, after Michael Somos *)

QP = QPochhammer; s = QP[q^2]^3*(QP[q^3]^2/QP[q]^2/QP[q^6]) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (Which[# < 2, 0^#2, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger@(6 n + 1))]; (* Michael Somos, Mar 05 2016 *)

PROG

(PARI) {a(n) = if( n<0, 0, sumdiv(6*n+1, d, kronecker(-3, d)))}; /* Michael Somos, Nov 03 2005 */

(PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 6*n+1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p>3, if( p%6==1, e+1, !(e%2)))))}; /* Michael Somos, Nov 03 2005 */

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^2 / (eta(x + A)^2 * eta(x^6 + A)), n))}; /* Michael Somos, Nov 03 2005 */

CROSSREFS

cf. A004016.

Sequence in context: A037809 A280534 A129451 * A274138 A179301 A008334

Adjacent sequences:  A097192 A097193 A097194 * A097196 A097197 A097198

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Sep 16 2004

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 14 10:20 EST 2019. Contains 329111 sequences. (Running on oeis4.)