A103440 a(n) = Sum[d|n, d==1 (mod 3), d^2] - Sum[d|n, d==2 (mod 3), d^2].

1, -3, 1, 13, -24, -3, 50, -51, 1, 72, -120, 13, 170, -150, -24, 205, -288, -3, 362, -312, 50, 360, -528, -51, 601, -510, 1, 650, -840, 72, 962, -819, -120, 864, -1200, 13, 1370, -1086, 170, 1224, -1680, -150, 1850, -1560, -24, 1584, -2208, 205, 2451, -1803, -288, 2210, -2808, -3, 2880, -2550

Offset

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..10000

G. E. Andrews and B. C. Berndt, Your Hit Parade: The Top Ten Most Fascinating Formulas in Ramanujan's Lost Notebook, Notices Amer. Math. Soc., 55 (No. 1, 2008), 18-30. See p. 23, Equation (27).

J. Stienstra, Mahler measure, Eisenstein series and dimers, arXiv:math/0502197 [math.NT], 2005.

Formula

G.f.: F(q) = Sum_{n>=1} A049347(n-1) * n^2 * q^n / (1 - q^n).

G.f.: F(q) = -q * G'(q) / (9*G(q)), with G(q) = Product_{n>=1} (1 - q^n)^(9*n * A049347(n-1)).

a(n) is multiplicative with a(3^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - z * a(p^(e-2)) where z = Kronecker(-3, p) * p^2 and a(p) = z + 1.

a(3*n) = a(n).

G.f.: Sum_{k>0} x^k * (1 - x^k - 6*x^(2*k) - x^(3*k) + x^(4*k)) / (1 + x^k + x^(2*k))^3. - Michael Somos, Oct 21 2007

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 - v + w + 3*v^2 - 8*w^2 + 6*v*w - 8*u*w + 6*u*v - 9*v^3 - 54*u*v*w + 72*u*w^2 - 9*u^2*w. - Michael Somos, Dec 23 2007

Example

G.f. = q - 3*q^2 + q^3 + 13*q^4 - 24*q^5 - 3*q^6 + 50*q^7 - 51*q^8 + q^9 + ...

Maple

f:= proc(n) local D, d;
  D:= numtheory:-divisors(n/3^padic:-ordp(n, 3));
  -add((-1)^(d mod 3)*d^2, d = D)
end proc:
map(f, [$1..100]); # Robert Israel, Aug 16 2018

Mathematica

a[n_] := Sum[m=Mod[d, 3]; (Boole[m==1]-Boole[m==2]) d^2, {d, Divisors[n]}];
Array[a, 56] (* Jean-François Alcover, Aug 16 2018 *)
a[ n_] := SeriesCoefficient[ (1 - QPochhammer[ x]^9 / QPochhammer[ x^3]^3) / 9, {x, 0, n}]; (* Michael Somos, Sep 07 2018 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d^2 * kronecker( -3, d)))}; /* Michael Somos, Oct 21 20007 */
(PARI) {a(n) = my(A, p, e, a0, a1, x, y, z); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 1, z = kronecker( -3, p) * p^2; a0 = 1; a1 = y = z + 1; for(i=2, e, x = y * a1 - z * a0; a0 = a1; a1 = x); a1)))}; /* Michael Somos, Oct 21 20007 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^9 / eta(x^3 + A)^3) / 9, n))}; /* Michael Somos, Oct 21 20007 */

Crossrefs

Equals A103637(n) - A103638(n). Cf. A002173.

A109041(n) = -9 * a(n) unless n=0. A014985(n) = a(2^n). -24 * A134340(n) = a(6*n+5).

Sequence in context: A331998 A053286 A008826 * A116483 A262593 A010290

Adjacent sequences: A103437 A103438 A103439 * A103441 A103442 A103443

Keyword

sign,mult

Author

Ralf Stephan, Feb 11 2005

Status

approved