A321527 -id:A321527 - cp's OEIS frontend

A224226 a(0)=1; thereafter a(n) =s(n,3)-s(n,4)-s(n,6)+s(n,12), where s(n,k) = sigma(n/k) if k divides n, otherwise 0.

1, 0, 0, 1, -1, 0, 2, 0, -3, 4, 0, 0, 1, 0, 0, 6, -7, 0, 8, 0, -6, 8, 0, 0, -1, 0, 0, 13, -8, 0, 12, 0, -15, 12, 0, 0, 7, 0, 0, 14, -18, 0, 16, 0, -12, 24, 0, 0, -5, 0, 0, 18, -14, 0, 26, 0, -24, 20, 0, 0, 6, 0, 0, 32, -31, 0, 24, 0, -18, 24, 0, 0, 5, 0, 0, 31, -20, 0, 28

Offset: 0

N. J. A. Sloane, Apr 09 2013

sign

Amiram Eldar, Table of n, a(n) for n = 0..10000
K. S. Williams, The parents of Jacobi's four squares theorem are unique, Amer. Math. Monthly, 120 (2013), 329-345.

Cf. A000203, A321527.

s := proc(n,k)
    if modp(n,k) = 0 then
        numtheory[sigma](n/k) ;
    else
        0 ;
    end if;
end proc:
A224226 := proc(n)
    if n = 0 then
        1;
    else
        s(n,3)-s(n,4)-s(n,6)+s(n,12) ;
    end if;
end proc: # R. J. Mathar, Nov 14 2018

s[n_, k_] := If[Divisible[n, k], DivisorSigma[1, n/k], 0]; a[0] = 1; a[n_] := s[n, 3] - s[n, 4] - s[n, 6] + s[n, 12]; Array[a, 100, 0] (* Amiram Eldar, Aug 17 2019 *)

s(n,k) = if (!(n%k), sigma(n/k), 0);
a(n) = if (n==0, 1, s(n,3)-s(n,4)-s(n,6)+s(n,12)); \\ Michel Marcus, Sep 27 2017

A321528 Expansion of b(x)^2 * b(x^2) / b(x^4) where b is a cubic AGM theta function.

Original entry on oeis.org

1, -6, 6, 30, -66, -36, 186, -48, -210, 138, 36, -72, 114, -84, 48, 180, -498, -108, 726, -120, -396, 240, 72, -144, -30, -186, 84, 462, -528, -180, 1116, -192, -1074, 360, 108, -288, 654, -228, 120, 420, -1260, -252, 1488, -264, -792, 828, 144, -288, -318

Offset: 0

Michael Somos, Nov 12 2018

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Number 64 of the 126 eta-quotients listed in Table 1 of Williams 2012.

			G.f. = 1 - 6*x + 6*x^2 + 30*x^3 - 66*x^4 - 36*x^5 + 186*x^6 - 48*x^7 + ...

Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

Cf. A098098, A134077, A321527.

A := Basis( ModularForms( Gamma0(12), 2), 49); A[1] - 6*A[2] + 6*A[3] + 30*A[4] - 66*A[5];

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^2])^3 / ( EllipticTheta[ 4, 0, x^3] EllipticTheta[ 4, 0, x^6]), {x, 0, n}];
a[ n_] := With[ {s = If[ FractionalPart @ # > 0, 0, DivisorSigma[1, #]] &}, If[ n < 1, Boole[n==0], -6 (s[n/1] - 4 s[n/2] - 9 s[n/3] + 16 s[n/4])]];
a[ n_] := If[ n < 1, Boole[n==0], -6 Sum[ d {1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0}[[Mod[d, 12, 1]]], {d, Divisors[n]}]];

{a(n) = if( n<1, n==0, -6 * sumdiv( n, d, d * [0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1][d%12 + 1]))};

{a(n) = my(s = x -> if(frac(x), 0, sigma(x))); if( n<1, n==0, -6 * (s(n/1) - 4*s(n/2) - 9*s(n/3) + 16*s(n/4)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^6 * eta(x^2 + A)^3 * eta(x^12 + A) / (eta(x^3 + A)^2 * eta(x^4 + A)^3 * eta(x^6 + A)), n))};

Expansion of phi(-x) * phi(-x^2)^3 / (phi(-x^3) * phi(-x^6)) in powers of x where phi() is a Ramanujan theta function.
Expansion of eta(q)^6 * eta(q^2)^3 * eta(q^12) / (eta(q^3)^2 * eta(q^4)^3 * eta(q^6)) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 864 (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A321527.
a(n) = -6 * (s(n/1) - 4*s(n/2) - 9*s(n/3) + 16*s(n/4)) if n>0, where s(x) = sum of divisors of x for integer x else 0.
a(2*n + 1) = -6 * A134077(n). a(6*n + 5) = -a(12*n + 10) = -36 * A098098(n).

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A224226 a(0)=1; thereafter a(n) =s(n,3)-s(n,4)-s(n,6)+s(n,12), where s(n,k) = sigma(n/k) if k divides n, otherwise 0.

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A321528 Expansion of b(x)^2 * b(x^2) / b(x^4) where b is a cubic AGM theta function.

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