A033687 -id:A033687 - cp's OEIS frontend

A375148 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+2)).

1, 1, 2, 1, 1, 1, 2, 1, 2, 0, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 0, 1, 1, 1, 3, 1, 2, 0, 2, 1, 2, 1, 1, 0, 2, 2, 2, 0, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 0, 2, 2, 1, 1, 0, 2, 0, 2, 1, 2, 1, 4, 1, 2, 0, 0, 1, 3, 1, 0, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 2, 0, 1, 0, 3, 2, 1, 1, 0, 2, 2, 1, 4, 0, 0

Offset: 0

Seiichi Manyama, Aug 01 2024

nonn

Cf. A374900, A375106, A375149, A375150.

Cf. A033687, A340453.

my(N=110, x='x+O('x^N)); Vec(sum(k=-N, N, x^k/(1-x^(7*k+2))))

my(N=110, x='x+O('x^N)); Vec(prod(k=1, N, (1-x^(7*k))^3*((1-x^(7*k-3))*(1-x^(7*k-4)))^2/(1-x^k)))

G.f.: Product_{k>0} (1-x^(7*k))^3 * ((1-x^(7*k-3)) * (1-x^(7*k-4)))^2 / (1-x^k).

G.f.: Sum_{k in Z} x^(2*k) / (1 - x^(7*k+1)).

A378007 Square table read by descending antidiagonals: T(n,k) = A378006(k*n+1,k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 0, 1, 0, 1, 1, 1, 2, 4, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 0, 2, 0, 0, 2, 1, 1, 1, 0, 4, 0, 1, 0, 3, 0, 1, 1, 1, 4, 6, 2, 6, 2, 4, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 1, 1, 4, 10, 4, 6, 4, 6, 2, 4, 2, 2, 1, 1

Offset: 0

Jianing Song, Nov 14 2024

A condensed version of A378006: the k-th column is the sequence {b(k*n+1)}, with the sequence {b(n)} having Dirichlet g.f. Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo k.

			Table starts
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 1, 2, 0, 2, 2, 2, 0, 4, ...
  1, 1, 2, 1, 4, 2, 0, 4, 6, 0, ...
  1, 1, 0, 2, 1, 2, 0, 2, 0, 4, ...
  1, 1, 2, 2, 0, 1, 6, 0, 6, 4, ...
  1, 1, 1, 0, 0, 2, 0, 4, 0, 0, ...
  1, 1, 2, 3, 4, 2, 6, 2, 0, 4, ...
  1, 1, 0, 2, 0, 2, 0, 0, 1, 4, ...
  1, 1, 1, 0, 4, 3, 0, 0, 6, 1, ...
  1, 1, 2, 2, 0, 0, 3, 4, 0, 0, ...
  1, 1, 2, 2, 0, 2, 6, 3, 0, 4, ...
Write w = exp(2*Pi*i/3) = (-1 + sqrt(3)*i)/2.
Column k = 1: 1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + 1/7^s + 1/8^s + 1/9^s + 1/10^s + 1/11^s + ...;
Column k = 2: 1 + 1/3^s + 1/5^s + 1/7^s + 1/9^s + 1/11^s + 1/13^s + 1/15^s + 1/17^s + 1/19^s + 1/21^s + ...;
Column k = 3: (1 + 1/2^s + 1/4^s + 1/5^s + ...)*(1 - 1/2^s + 1/4^s - 1/5^s + ...) = 1 + 1/4^s + 2/7^s + 2/13^s + 1/16^s + 2/19^s + 1/25^s + 2/28^s + 2/31^s + ...;
Column k = 4: (1 + 1/3^s + 1/5^s + 1/7^s + ...)*(1 - 1/3^s + 1/5^s - 1/7^s + ...) = 1 + 2/5^s + 1/9^s + 2/13^s + 2/17^s + 3/25^s + 2/29^s + 2/37^s + 2/41^s + ...;
Column k = 5: (1 + 1/2^s + 1/3^s + 1/4^s + ...)*(1 + i/2^s - i/3^s - 1/4^s + ...)*(1 - 1/2^s - 1/3^s + 1/4^s + ...)*(1 - i/2^s + i/3^s - 1/4^s + ...) = 1 + 4/11^s + 1/16^s + 4/31^s + 4/41^s + ...;
Column k = 6: (1 + 1/5^s + 1/7^s + 1/11^s + ...)*(1 - 1/5^s + 1/7^s - 1/11^s + ...) = 1 + 2/7^s + 2/13^s + 2/19^s + 1/25^s + 1/31^s + 2/37^s + 2/43^s + 3/49^s + 2/61^s + ...;
Column k = 7: (1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + ...)*(1 + w/2^s + (w+1)/3^s - (w+1)/4^s - w/5^s - 1/6^s + ...)*(1 - (w+1)/2^s + w/3^s + w/4^s - (w+1)/5^s + 1/6^s + ...)*(1 + 1/2^s - 1/3^s + 1/4^s - 1/5^s - 1/6^s + ...)*(1 + w/2^s - (w+1)/3^s - (w+1)/4^s + w/5^s + 1/6^s + ...)*(1 - (w+1)/2^s - w/3^s + w/4^s + (w+1)/5^s - 1/6^s + ...) = 1 + 2/8^s + 6/29^s + 6/43^s + 3/64^s + 6/71^s + ...;
Column k = 8: (1 + 1/3^s + 1/5^s + 1/7^s + ...)*(1 + 1/3^s - 1/5^s - 1/7^s + ...)*(1 - 1/3^s + 1/5^s - 1/7^s + ...)*(1 - 1/3^s - 1/5^s + 1/7^s + ...) = 1 + 2/9^s + 4/17^s + 2/25^s + 4/41^s + 2/49^s + 4/73^s + 3/81^s + ...;
Column k = 9: (1 + 1/2^s + 1/4^s + 1/5^s + 1/7^s + 1/8^s + ...)*(1 + (w+1)/2^s + w/4^s - w/5^s - (w+1)/7^s - 1/8^s + ...)*(1 + w/2^s - (w+1)/4^s - (w+1)/5^s + w/7^s + 1/8^s + ...)*(1 - 1/2^s + 1/4^s - 1/5^s + 1/7^s - 1/8^s + ...)*(1 - (w+1)/2^s + w/4^s + w/5^s - (w+1)/7^s + 1/8^s + ...)*(1 - w/2^s - (w+1)/4^s + (w+1)/5^s + w/7^s - 1/8^s + ...) = 1 + 6/19^s + 6/37^s + 1/64^s + 6/73^s + ...;
Column k = 10: (1 + 1/3^s + 1/7^s + 1/9^s + ...)*(1 + i/3^s - i/7^s - 1/9^s + ...)*(1 - 1/3^s - 1/7^s + 1/9^s + ...)*(1 - i/3^s + i/7^s - 1/9^s + ...) = 1 + 4/11^s + 4/31^s + 4/41^s + 4/61^s + 4/71^s + 1/81^s + 4/101^s + ...

Jianing Song, Table of n, a(n) for n = 0..11324 (the first 150 diagonals, with n+k = 1..150)

Columns: A000012 (k=1 and k=2), A033687 (k=3), A008441 (k=4), A378008 (k=5), A097195 (k=6), A378009 (k=7), A378010 (k=8), A378011 (k=9), A378012 (k=10).

Cf. A378006.

A378007(n,k) = {
my(f = factor(k*n+1), res = 1); for(i=1, #f~, my(d = znorder(Mod(f[i,1],k)));
if(f[i,2] % d != 0, return(0), my(m = f[i,2]/d, r = eulerphi(k)/d); res *= binomial(m+r-1,r-1)));
res;}

See A378006.

For odd k, T(2*k,n) = T(k,2*n).

A045833 Expansion of eta(q^9)^3 / eta(q^3) in powers of q.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0

Offset: 0

N. J. A. Sloane

			G.f. = q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + q^25 + 2*q^28 + 2*q^31 + ...

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A033685, A033687, A097195, A217219.

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^9]^3 / QPochhammer[ q^3], {q, 0, n}]; (* Michael Somos, Feb 22 2015 *)
f[p_, e_] := If[Mod[p, 3] == 1, e + 1, (1 + (-1)^e)/2]; f[3, e_] := 0; a[0] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* Amiram Eldar, Oct 13 2022 *)

{a(n) = local(A, p, e); if( n<0, 0, A=factor(n); prod(k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p!=3, if( p%3==1, e+1, !(e%2))))))}; \\ Michael Somos, May 25 2005

{a(n) = local(A); if( (n<1) || (n%3!=1), 0, n = (n-1)/3; A = x * O(x^n); polcoeff( eta(x^3 + A)^3 / eta(x + A), n))}; \\ Michael Somos, May 25 2005

From Michael Somos, May 25 2005: (Start)
Euler transform of period 9 sequence [ 0, 0, 1, 0, 0, 1, 0, 0, -2, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2*w - 2*u*w^2 - v^3.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1*u3^2 + u1*u6^2 - u1*u3*u6 - u2^2*u3.
a(3*n) = a(3*n + 2) = 0. a(3*n + 1) = A033687(n). a(6*n + 1) = A097195(n). 3*a(n) = A033685(n).
Multiplicative with a(3^e) = 0^e, a(p^e) = e+1 if p == 1 (mod 3), a(p^e) = (1+(-1)^e)/2 if p == 2 (mod 3).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u2*u3^2 + 2*u2*u3*u6 + 4*u2*u6^2 - u1^2*u6. (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(9*sqrt(3)) = 0.403066... . - Amiram Eldar, Oct 13 2022
Dirichlet g.f.: L(chi_1,s)*L(chi_{-1},s), where chi_1 = A011655 and chi_{-1} = A102283 are respectively the principal and the non-principal Dirichlet character modulo 3. For the formula of the sequence whose Dirichlet g.f. is Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo k, see A378006. This sequence is the case k = 3. - Jianing Song, Nov 13 2024

A106402 Expansion of eta(q^3)^9 / eta(q)^3 in powers of q.

Original entry on oeis.org

1, 3, 9, 13, 24, 27, 50, 51, 81, 72, 120, 117, 170, 150, 216, 205, 288, 243, 362, 312, 450, 360, 528, 459, 601, 510, 729, 650, 840, 648, 962, 819, 1080, 864, 1200, 1053, 1370, 1086, 1530, 1224, 1680, 1350, 1850, 1560, 1944, 1584, 2208, 1845, 2451, 1803, 2592

Offset: 1

Michael Somos, May 02 2005

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Number 3 of the 74 eta-quotients listed in Table I of Martin (1996).
a(n+1) is the number of partition triples of n where each partition is 3-core (see Theorem 3.1 of Wang link).
Convolution cube of A033687.
Convolution square is A198958. - Michael Somos, Dec 26 2015

			G.f. = q + 3*q^2 + 9*q^3 + 13*q^4 + 24*q^5 + 27*q^6 + 50*q^7 + 51*q^8 + ...

George E. Andrews and Bruce C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001). See p. 314, Eq. (14.2.14).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vaclav Kotesovec)
Jonathan M. Borwein and Peter B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012). See page 697.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Hossein Movasati and Younes Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv preprint arXiv:1603.09411 [math.AG], 2016-2021.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
Liuquan Wang, Explicit Formulas for Partition Pairs and Triples with 3-Cores, arXiv:1507.03099 [math.NT], 2015.

Cf. A004016, A005882, A005928, A033687, A109041, A129404, A198958, A231947.

A := Basis( ModularForms( Gamma1(3), 3), 52); A[2]; /* Michael Somos, May 18 2015 */

a[ n_] := If[ n < 1, 0, DivisorSum[ n, #^2 KroneckerSymbol[ n/#, 3] &]]; (* Michael Somos, Jul 19 2012 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^3]^3 / QPochhammer[ q])^3, {q, 0, n}]; (* Michael Somos, Jul 19 2012 *)
nmax = 40; Rest[CoefficientList[Series[x * Product[(1 - x^(3*k))^9 / (1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 07 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A)^9 / eta(x + A)^3, n))};

{a(n) = if( n<1, 0, sumdiv( n, d, d^2 * kronecker( n/d, 3)))};

{a(n) = my(A, p, e, u); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; u = kronecker(-3, p); ((p^2)^(e+1) - u^(e+1)) / (p^2 - u)))};

a(n) = sumdiv(n, d, ((d % 3) == 1)*(n/d)^2) - sumdiv(n, d, ((d % 3)== 2)*(n/d)^2); \\ Michel Marcus, Jul 14 2015

Expansion of (c(q) / 3)^3 in powers of q where c(q) is a cubic AGM theta function.
Euler transform of period 3 sequence [ 3, 3, -6, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + 6*u*v*w + 8*u*w^2 - u^2*w.
G.f.: Sum_{k>0} k^2 * x^k / (1 + x^k + x^(2*k)) = x * Product_{k>0} (1 - x^(3*k))^9 / (1 - x^k)^3.
a(n) is multiplicative and a(p^e) = ((p^2)^(e+1) - u^(e+1)) / (p^2 - u) where u = 0, 1, -1 when p == 0, 1, 2 (mod 3). - Michael Somos, Oct 19 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 27^(-1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A109041.
a(3*n) = 9 * a(n). a(3*n + 1) = A231947(n). -  Michael Somos, May 18 2015
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 4*Pi^3/(81*sqrt(3)) = 0.8840238... (A129404). - Amiram Eldar, Nov 09 2023

A374900 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+1)).

Original entry on oeis.org

1, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 1, 4, 2, 2, 2, 0, 2, 2, 3, 4, 2, 0, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 0, 2, 2, 0, 2, 2, 2, 3, 2, 0, 2, 2, 4, 0, 2, 2, 2, 2, 2, 3, 2, 0, 2, 4, 2, 2, 0, 0, 2, 2, 2, 4, 2, 0, 2, 2, 2, 2, 2, 0, 2, 4, 4, 2, 2, 0, 2, 2, 2, 2, 2

Offset: 0

Seiichi Manyama, Jul 31 2024

nonn

Cf. A008441, A033687, A097195, A340456.

my(N=110, x='x+O('x^N)); Vec(sum(k=-N, N, x^k/(1-x^(7*k+1))))

my(N=110, x='x+O('x^N)); Vec(prod(k=1, N, (1-x^(7*k))^2*(1-x^(7*k-2))*(1-x^(7*k-5))/((1-x^(7*k-1))*(1-x^(7*k-6)))^2))

G.f.: Product_{k>0} (1-x^(7*k))^2 * (1-x^(7*k-2)) * (1-x^(7*k-5)) / ((1-x^(7*k-1)) * (1-x^(7*k-6)))^2.

A123477 Expansion of (1 - b(q)) / 3 in powers of q where b(q) is a cubic AGM theta function.

Original entry on oeis.org

1, 0, -2, 1, 0, 0, 2, 0, -2, 0, 0, -2, 2, 0, 0, 1, 0, 0, 2, 0, -4, 0, 0, 0, 1, 0, -2, 2, 0, 0, 2, 0, 0, 0, 0, -2, 2, 0, -4, 0, 0, 0, 2, 0, 0, 0, 0, -2, 3, 0, 0, 2, 0, 0, 0, 0, -4, 0, 0, 0, 2, 0, -4, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, -2, 2, 0, 0, 2, 0, -2, 0, 0, -4, 0, 0, 0, 0, 0, 0, 4, 0, -4, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0

Offset: 1

Michael Somos, Sep 27 2006

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Denoted by lambda(n) on page 4 (1.7) in Kassel and Reutenauer arXiv:1610.07793. - Michael Somos, Dec 10 2017

			G.f. = q - 2*q^3 + q^4 + 2*q^7 - 2*q^9 - 2*q^12 + 2*q^13 + q^16 + 2*q^19 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv preprint arXiv:1610.07793 [math.NT], 2016.

Cf. A002324, A005928, A033687, A113063.

A123477 := proc(n)
    local a,pe,p,e;
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if modp(p,6) = 1 then
            a := a*(e+1) ;
        elif modp(p,6) in {2,5} then
            a := a*(1+(-1)^e)/2 ;
        elif e > 0 then
            a := -2*a ;
        end if;
    end do:
    a ;
end proc:
seq(A123477(n),n=1..100) ; # R. J. Mathar, Feb 22 2021

a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -1, -3, 1, -1, 3, 1, -1, 0} [[Mod[#, 9, 1]]] &]]; (* Michael Somos, Dec 10 2017 *)

{a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, -1, -3, 1, -1, 3, 1, -1] [d%9+1]))};

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, -2, p%6==1, e+1, !(e%2))))};

Moebius transform is period 9 sequence [1, -1, -3, 1, -1, 3, 1, -1, 0, ...].
a(n) is multiplicative and a(p^e) = -2 if p = 3 and e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 2, 5 (mod 6).
a(3*n + 2) = 0. a(3*n + 1) = A033687(n), a(3*n) = -2*A002324(n).
-3*a(n) = A005928(n) unless n=0. |a(n)| = A113063(n).

A130539 Expansion of q^(-1/3) * a(q) * b(q) * c(q) / 3 in powers of q where a(), b(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, 4, -13, 0, -1, 16, 11, 0, 25, -52, -46, 0, 47, 0, -22, 0, 120, -4, 0, 0, -121, 64, -109, 0, -97, 44, 131, 0, 0, 0, 13, 0, 167, 100, -37, 0, -214, -208, 0, 0, 121, -184, 146, 0, -143, 0, 251, 0, 0, 188, 59, 0, -118, 0, 299, 0, -168, -88, -325, 0, -313

Offset: 0

Michael Somos, Jun 03 2007

sign

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Denoted by g_3(q) in Cynk and Hulek in Remark 3.4 on page 12 as the unique level 27 form of weight 3.
This is a member of an infinite family of integer weight modular forms. g_1 = A033687, g_2 = A030206, g_3 = A130539, g_4 = A000731.
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).

			G.f. = 1 + 4*x - 13*x^2 - x^4 + 16*x^5 + 11*x^6 + 25*x^8 - 52*x^9 - 46*x^10 + ...
G.f. = q + 4*q^4 - 13*q^7 - q^13 + 16*q^16 + 11*q^19 + 25*q^25 - 52*q^28 - ...

S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds

Cf. A000731, A030206, A033687.

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^3] (QPochhammer[ x]^3 + 9 x QPochhammer[ x^9]^3), {x, 0, n}]; (* Michael Somos, Oct 20 2015 *)

{a(n) = my(A, p, e, x, y, a0, a1); n = 3*n + 1; if( n<1, 0, A=factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, if( p%3==2, if( e%2, 0, p^e), for( x=1, sqrtint(4*p\27), if( issquare(4*p - 27*x^2, &y), break)); y = y^2 - p*2; a0=1; a1=y; for( i=2, e, x=y*a1 - p^2*a0; a0=a1; a1=x); a1))))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^3 + A) * (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3), n))};

Expansion of q^(-1/3) * ( eta(q)^5 * eta(q^3) + 9 * eta(q)^2 * eta(q^3) * eta(q^9)^3 ) in powers of q.
a(n) = b(3*n + 1) where b() is multiplicative and b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^e if p == 2 (mod 3), b(p^e) = b(p) * b(p^(e-1)) - p^2 * b(p^(e-2)) if p == 1 (mod 3) where b(p) = x^2 - 2*p, 4*p = x^2 + 3*y^2, |x|<|y| and x == 2 (mod 3).
G.f.: Sum_{k>=0} a(k) * x^(3*k + 1) = (1/2) * Sum_{u, v in Z} (u*u - 7*v*v) * x^(u*u + u*v + 7*v*v). - Michael Somos, Jun 14 2007
a(4*n + 1) = 4*a(n). a(4*n + 3) = 0. - Michael Somos, Oct 20 2015

A261426 Expansion of f(-x^3)^3 * phi(x^6) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 2, 0, 2, 1, 4, 2, 5, 2, 6, 2, 6, 0, 4, 4, 7, 2, 4, 0, 6, 1, 8, 4, 4, 4, 10, 2, 8, 2, 12, 4, 8, 5, 6, 0, 14, 2, 8, 2, 11, 6, 6, 4, 8, 2, 8, 4, 8, 6, 14, 0, 6, 0, 12, 6, 15, 4, 14, 2, 14, 4, 8, 8, 12, 7, 14, 0, 12, 2, 16, 10, 8, 4, 10, 6, 14, 0, 16, 4, 16

Offset: 0

Michael Somos, Aug 18 2015

nonn

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).

			G.f. = 1 + x + 2*x^2 + 2*x^4 + x^5 + 4*x^6 + 2*x^7 + 5*x^8 + 2*x^9 + ...
G.f. = q + q^4 + 2*q^7 + 2*q^13 + q^16 + 4*q^19 + 2*q^22 + 5*q^25 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A033687, A261394, A261426, A261444.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 EllipticTheta[ 3, 0, x^6] / QPochhammer[ x], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^12 + A)^5 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^24 + A)^2), n))};

Expansion of (1/3) * q^(-1/3) * c(q) * phi(q^6) in powers of q where phi() is a Ramanujan theta function and c() is a cubic AGM function. - Michael Somos, Sep 01 2015
Expansion of q^(-1/3) * eta(q^3)^3 * eta(q^12)^5 / (eta(q) * eta(q^6)^2 * eta(q^24)^2) in powers of q.
Euler transform of period 24 sequence [ 1, 1, -2, 1, 1, 0, 1, 1, -2, 1, 1, -5, 1, 1, -2, 1, 1, 0, 1, 1, -2, 1, 1, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = (128/3)^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261426.
24 * a(n) = A261394(6*n + 2).
a(n) = A261444(2*n). Michael Somos, Sep 01 2015

A081622 Number of 6-core partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 5, 9, 10, 12, 12, 14, 20, 20, 21, 23, 24, 24, 32, 29, 35, 36, 44, 47, 38, 47, 49, 52, 55, 58, 59, 64, 66, 71, 70, 78, 79, 88, 87, 90, 85, 87, 111, 104, 102, 107, 112, 113, 121, 113, 130, 130, 148, 153, 132, 147, 149, 156, 162, 149, 167, 160, 178, 180

Offset: 0

Michael Somos, Mar 24 2003

Euler transform of period 6 sequence [ 1, 1, 1, 1, 1, -5, ...].

Expansion of q^(-35/24) * eta(q^6)^6 / eta(q) in powers of q.

			1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 5*x^6 + 9*x^7 + 10*x^8 + 12*x^9 + ...
q^35 + q^59 + 2*q^83 + 3*q^107 + 5*q^131 + 7*q^155 + 5*q^179 + 9*q^203 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
F. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17.

Cf. A010054, A033687, A045831, A053723, A053724.

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^(6*k) + x * O(x^n))^6 / (1 - x^k)), n))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^6 / eta(x + A), n))}

G.f.: Product_{k>0} (1 - x^(6*k))^6 / (1 - x^k).

A112298 Expansion of (a(q) - 3a(q^2) + 2a(q^4)) / 6 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

1, -3, 1, 3, 0, -3, 2, -3, 1, 0, 0, 3, 2, -6, 0, 3, 0, -3, 2, 0, 2, 0, 0, -3, 1, -6, 1, 6, 0, 0, 2, -3, 0, 0, 0, 3, 2, -6, 2, 0, 0, -6, 2, 0, 0, 0, 0, 3, 3, -3, 0, 6, 0, -3, 0, -6, 2, 0, 0, 0, 2, -6, 2, 3, 0, 0, 2, 0, 0, 0, 0, -3, 2, -6, 1, 6, 0, -6, 2, 0, 1, 0, 0, 6, 0, -6, 0, 0, 0, 0, 4, 0, 2, 0, 0, -3, 2, -9, 0, 3, 0, 0, 2, -6, 0

Offset: 1

Michael Somos, Sep 02 2005

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

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

			G.f. = q - 3*q^2 + q^3 + 3*q^4 - 3*q^6 + 2*q^7 - 3*q^8 + q^9 + 3*q^12 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A033762, A093602, A093829, A097195, A112604, A112605, A129576, A244375.

Cf. A000122, A000700, A004016, A010054, A005882, A005928, A121373.

A := Basis( ModularForms( Gamma1(12), 1), 106); A[2] - 3*A[3] + A[4] + 3*A[5]; /* Michael Somos, Jan 17 2015 */

a[ n_] := SeriesCoefficient[ q QPochhammer[ q, q^2]^3 QPochhammer[ -q^6, q^6]^3 EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, q^(3/2)] / (2 q^(3/8)), {q, 0, n}]; (* Michael Somos, Jan 17 2015 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, JacobiSymbol[ -3, n/#] {1, -2, 1, 0}[[Mod[#, 4, 1]]] &]]; (* Michael Somos, Jan 17 2015 *)

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

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^12 + A))^3/ (eta(x^2 + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^6 + A)), n))};

From Michael Somos, Jan 17 2015: (Start)
Expansion of b(q) * (b(q^4) - b(q)) / (3*b(q^2)) in powers of q  where b() is a cubic AGM theta function.
Expansion of q * chi(-q)^3 * phi(-q^2) * psi(q^3) / chi(-q^6)^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q * phi(-q)^2 * psi(q^6)^2 / (psi(-q) * psi(-q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of q * f(q) * f(-q, -q^5)^4 / f(q^3)^3 in powers of q where f() is a Ramanujan theta function. (End)
Expansion of (eta(q) * eta(q^12))^3 / (eta(q^2) * eta(q^3) * eta(q^4) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ -3, -2, -2, -1, -3, 0, -3, -1, -2, -2, -3, -2, ...].
Moebius transform is period 12 sequence [ 1, -4, 0, 6, -1, 0, 1, -6, 0, 4, -1, 0, ...].
Multiplicative with a(2^e) = 3(-1)^e if e>0, a(3^e)=1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 2 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Sum_{k>0} Kronecker(-3, k) * x^k * (1 - x^k)^2 / (1 - x^(4*k)).
a(n) = -(-1)^n * A244375(n). a(6*n + 5) = 0, a(3*n) = a(n).
a(2*n) = -3 * A093829(n). a(2*n + 1) = A033762(n). a(3*n + 1) = A129576(n). a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n). a(6*n + 1) = A097195(n). a(6*n + 2) = -3 * A033687(n).
Sum_{k=1..n} abs(a(k)) ~ (Pi/sqrt(3)) * n. - Amiram Eldar, Jan 23 2024

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A378007 Square table read by descending antidiagonals: T(n,k) = A378006(k*n+1,k).

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A106402 Expansion of eta(q^3)^9 / eta(q)^3 in powers of q.

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A374900 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+1)).

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A123477 Expansion of (1 - b(q)) / 3 in powers of q where b(q) is a cubic AGM theta function.

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Maple

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A130539 Expansion of q^(-1/3) * a(q) * b(q) * c(q) / 3 in powers of q where a(), b(), c() are cubic AGM theta functions.

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A112298 Expansion of (a(q) - 3a(q^2) + 2a(q^4)) / 6 in powers of q where a() is a cubic AGM theta function.