A033687 -id:A033687 - cp's OEIS frontend

A112848 Expansion of eta(q)eta(q^2)eta(q^18)^2/(eta(q^6)*eta(q^9)) in powers of q.

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

Offset: 1

Michael Somos, Sep 22 2005

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A033687, A033762, A092829, A093829, A097195, A248897, A255648 (absolute values).

QP = QPochhammer; s = QP[q]*QP[q^2]*(QP[q^18]^2/(QP[q^6]*QP[q^9])) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)
f[p_, e_] := If[Mod[p, 6] == 1, e+1, (1+(-1)^e)/2]; f[2, e_] := (-1)^e; f[3, e_]:= -2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 28 2024 *)

{a(n)=if(n<1, 0, if(n%3==0, n/=3; -2,1)* sumdiv(n,d,kronecker(-12,d) -if(d%2==0, 2*kronecker(-3,d/2))))}

{a(n)=local(A); if (n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)*eta(x^2+A)*eta(x^18+A)^2/ eta(x^6+A)/eta(x^9+A), n))}

Euler transform of period 18 sequence [ -1, -2, -1, -2, -1, -1, -1, -2, 0, -2, -1, -1, -1, -2, -1, -2, -1, -2, ...].
Moebius transform is period 18 sequence [1, -2, -3, 2, -1, 6, 1, -2, 0, 2, -1, -6, 1, -2, 3, 2, -1, 0, ...].
Multiplicative with a(2^e) = (-1)^e, a(3^e) = -2 if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} Kronecker(-3, k) x^k(1-x^(2k))^2/(1-x^(6k)) = x Product_{k>0} (1-x^k)(1-x^(2k))(1+x^(9k))(1+x^(6k)+x^(12k)).
a(3n) = -2*A092829(n). a(3n+1) = A093829(3n+1) = A033687(n). a(3n+2) = A093829(3n+2). a(6n)/2 = A093829(n). a(6n+1) = A097195(n). a(6n+3) = -2*A033762(n). a(6n+5) = 0.
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Jan 23 2024

A113448 Expansion of (eta(q^2)^2 * eta(q^9) * eta(q^18)) / (eta(q) * eta(q^6)) in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Nov 02 2005

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

			G.f. = x + x^2 + x^4 + 2*x^7 + x^8 + 2*x^13 + 2*x^14 + x^16 + 2*x^19 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A004016, A005882, A005928, A033687, A073010, A097195.

a[ n_] := If[ n < 1, 0, If[ Mod[n, 3] == 0, 0, DivisorSum[ n, KroneckerSymbol[ -12, #] &]]]; (* Michael Somos, Jul 30 2015 *)
a[ n_] := SeriesCoefficient[ x QPochhammer[ x^9]^3 / QPochhammer[ x^3] + x^2 QPochhammer[ x^18]^3 / QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Jul 30 2015 *)

{a(n) = if( n<1, 0, if( n%3, sumdiv(n,d, kronecker(-12, d))))};

{a(n) = if( n<1, 0, direuler(p=2, n, if( p==3, 1, 1 / ((1 - X) * (1 - kronecker(-12, p)*X))))[n])}

{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==2, 1, p==3, 0, p%6==1, e+1, !(e%2))))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^9 + A) * eta(x^18 + A) / (eta(x + A) * eta(x^6 + A)), n))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^9 + A)^3 / eta(x^3 + A) + x * eta(x^18 + A)^3 / eta(x^6 + A), n))};

Euler transform of period 18 sequence [ 1, -1, 1, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 1, -1, 1, -2, ...].
Moebius transform is period 18 sequence [ 1, 0, -1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 1, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 0^e, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
a(3*n) = 0, a(2*n) = a(n).
G.f.: Sum_{k>0} x^(6*k - 5) / (1 - x^(6*k - 5)) - x^(6*k - 1) / (1 - x^(6*k - 1)) - x^(18*k - 15) / (1 - x^(18*k - 15)) + x^(18*k - 6) / (1 - x^(18*k - 6)).
G.f.: Sum_{k>0} x^k * (1 - x^(2*k)) * (1 - x^(4*k)) * (1-x^(10*k)) / (1 - x^(18*k)).
Expansion of (c(q) + c(q^2))/3 in powers of q^(1/3) where c(q) is a cubic AGM theta function.
a(3*n + 1) = A033687(n). a(6*n + 1) = A097195(n). - Michael Somos, Jul 30 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Oct 15 2022

A138806 Expansion of (theta_3(q) * theta_3(q^27) + theta_2(q) * theta_2(q^27) - 1) / 2 in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Mar 30 2008

Half the number of integer solutions to x^2 + x*y + 7*y^2 = n. - Jianing Song, Nov 20 2019

			q + q^4 + 2*q^7 + 3*q^9 + 2*q^13 + q^16 + 2*q^19 + q^25 + 3*q^27 + ...

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A138805 (number of integer solutions to x^2 + x*y + 7*y^2 = n).
Cf. A033687, A073010, A097195, A123884, A121361.
Similar sequences: A096936, A113406, A110399.

f[p_, e_] := If[Mod[p, 6] == 1, e + 1, (1 + (-1)^e)/2]; f[2, e_] := 1 - Mod[e, 2]; f[3, e_] := 3; f[3, 1] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 07 2023 *)

{a(n) = if( n<1, 0, if( n%3 == 2, 0, if( n%3==1, sumdiv(n, d, kronecker(-3, d)), if( n%9==0, 3 * sumdiv(n/9, d, kronecker(-3, d))))))}

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

{a(n) = if( n<1, 0, qfrep([2, 1; 1, 14], n, 1)[n])}

a(n) is multiplicative and a(3^e) = 3 if e>1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
a(3*n + 2) = a(4*n + 2) = 0.
G.f.: (Sum_{i,j} x^(i*i + i*j + 7*j*j) - 1) / 2.
A138805(n) = 2 * a(n) unless n=0. A033687(n) = a(3*n + 1). A097195(n) = a(6*n + 1). A123884(n) = a(12*n + 1). 2 * A121361(n) = a(12*n + 7).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Nov 16 2023

A244375 Expansion of (a(q) + 3a(q^2) - 4a(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

Offset: 1

Michael Somos, Jun 26 2014

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. = 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, A112298, A122861, A244339.

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

A := Basis( ModularForms( Gamma1(12), 1), 82); A[2] + 3*A[3] + A[4] - 3*A[5]; /* Michael Somos, Jan 17 2015 */

a[ n_] := If[ n < 1, 0, Sum[ Mod[ n/d, 2] {1, 3, 0, -3, -1, 0}[[ Mod[ d, 6, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^8 QPochhammer[ q^3] QPochhammer[ q^12]^4 / (QPochhammer[ q]^3 QPochhammer[ q^4]^4 QPochhammer[ q^6]^4), {q, 0, n}];
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q, q^2]^3 QPochhammer[ -q^6, q^6]^3 EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, Pi/4, q^(3/2)] / (2^(1/2) q^(3/8)), {q, 0, n}]; (* Michael Somos, Jan 17 2015 *)

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

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^3 + A) * eta(x^12 + A)^4 / (eta(x + A)^3 * eta(x^4 + A)^4 * eta(x^6 + A)^4), n))};

{a(n) = if( n<0, 0, polcoeff( sum(k=1, n, x^k / (1 + x^k + x^(2*k)) * [0, 1, 4, 1][k%4 + 1], x * O(x^n)), n))};

{a(n) = if( n<0, 0, polcoeff( sum(k=1, n, x^k / (1 - x^(2*k)) * [0, 1, 3, 0, -3, -1][k%6 + 1], x * O(x^n)), n))};

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

Expansion of (b(q) - b(q^4)) * (b(q) - 2*b(q^4)) / (3* b(q^2)) = b(q^2)^2 * (b(q^4) - b(q)) / (3 * b(q) * b(q^4)) in powers of q where b() is a cubic AGM theta function.
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 * 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. - Michael Somos, Jan 17 2015
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. - Michael Somos, Jan 17 2015
Expansion of eta(q^2)^8 * eta(q^3) * eta(q^12)^4 / (eta(q)^3 * eta(q^4)^4 * eta(q^6)^4) in powers of q. - Michael Somos, Jan 17 2015
a(n) is multiplicative with a(2^e) = 3 * (-1)^(e+1) 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 == 5 (mod 6).
Euler transform of period 12 sequence [ 3, -5, 2, -1, 3, -2, 3, -1, 2, -5, 3, -2, ...].
Moebius transform is period 12 sequence [ 1, 2, 0, -6, -1, 0, 1, 6, 0, -2, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A244339.
a(2*n) = 3 * A093829(n). a(2*n + 1) = A033762(n). a(3*n) = a(n). a(3*n + 1) = A122861(n). a(6*n + 2) = 3 * A033687(n). a(6*n + 5) = 0.
a(n) = -(-1)^n * A112298(n). - Michael Somos, Jan 17 2015
Sum_{k=1..n} abs(a(k)) ~ (Pi/sqrt(3)) * n. - Amiram Eldar, Jan 23 2024

A113063 Associated with theta series of hexagonal net with respect to a node.

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, Oct 13 2005

Denoted by |lambda(n)| on page 4 (1.7) in Kassel and Reutenauer arXiv:1610.07793. - Michael Somos, Jun 04 2015

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

Antti Karttunen, Table of n, a(n) for n = 1..16384
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 5.

Cf. A002324, A033687, A113062, A121839, A123477.

a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -1, 1, 1, -1, -1, 1, -1, 0} [[Mod[#, 9, 1]]] &]]; (* Michael Somos, Jun 04 2015 *)
f[p_, e_] := If[Mod[p, 6] == 1, e+1, (1+(-1)^e)/2]; f[3, e_] := 2; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 05 2023 *)

{a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, -1, 1, 1, -1, -1, 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, 1, 1, -1, -1, 1, -1, 0, ...].
a(n) is multiplicative with 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) = A113062(n) unless n=0.
G.f.: Sum_{k>0} f(x^k) + f(x^(3*k)) where f(x) := x / (1 + x + x^2). - Michael Somos, Jun 04 2015
a(n) = |A123477(n)|. - Michael Somos, Dec 10 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/(9*sqrt(3)) = 0.806133... (A121839 - 1). - Amiram Eldar, Dec 28 2023

A123863 Expansion of (c(q^3) - c(q^6) - 2*c(q^12)) / 3 in powers of q where c(q) is a cubic AGM theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Oct 14 2006

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. = q - q^2 - q^4 + 2*q^7 - q^8 + 2*q^13 - 2*q^14 - q^16 + 2*q^19 + ...

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

Cf. A033687, A097195, A113448, A121361, A123884.

a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 4, {1, -1, 0}[[Mod[#, 3, 1]]], Mod[#, 6] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger @ n)]; (* Michael Somos, Aug 03 2015 *)
a[ n_] := SeriesCoefficient[ x EllipticTheta[ 2, 0, x^(9/2)] EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 4, 0, x^18] / (2^(3/2) x^(5/4) QPochhammer[ x^6]), {x, 0, n}]; (* Michael Somos, Aug 03 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^18 + A)^4 / (eta(x^2 + A) * eta(x^6 + A) * eta(x^9 + A) * eta(x^36 + A)), n))};

{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==2, -1, p==3, 0, p%6==1, e+1, !(e%2))))};

Expansion of (a(x) - a(x^2) - a(x^3) - 2*a(x^4) + a(x^6) + 2*a(x^12)) / 6 in powers of x where a() is a cubic AGM theta function. - Michael Somos, Aug 03 2015
Expansion of psi(-x) * psi(-x^9) * phi(x^9) / f(-x^6) in powers of x where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Aug 03 2015
Expansion of eta(q) * eta(q^4) * eta(q^18)^4 / (eta(q^2) * eta(q^6) * eta(q^9) * eta(q^36)) in powers of q.
Euler transform of period 36 sequence [ -1, 0, -1, -1, -1, 1, -1, -1, 0, 0, -1, 0, -1, 0, -1, -1, -1, -2, -1, -1, -1, 0, -1, 0, -1, 0, 0, -1, -1, 1, -1, -1, -1, 0, -1, -2, ...].
a(n) is multiplicative with a(2^e) = -1 if e>0, a(3^e) = 0^e, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
a(3*n) = a(6*n + 5) = 0.
a(2*n) = -A113448(n). a(6*n + 2) = -A033687(n).
a(3*n + 1) = A227696(n). a(6*n + 1) = A097195(n). a(12*n + 1) = A123884(n). a(12*n + 7) = 2 * A121361(n). - Michael Somos, Aug 03 2015

A182805 Number of 10-core partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 32, 46, 57, 71, 85, 106, 121, 147, 165, 190, 242, 267, 302, 350, 400, 443, 511, 565, 638, 715, 774, 852, 964, 1038, 1135, 1253, 1372, 1482, 1650, 1785, 1878, 2098, 2234, 2411, 2625, 2819, 2963, 3249, 3393, 3600, 4004, 4181

Offset: 0

Alois P. Heinz, Dec 03 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

10th column of A175595.

Cf. A033687, A045831, A053723, A081622, A053724, A182803, A182804.

with(numtheory): A:= proc(n, t) option remember; local d, j; `if`(n=0, 1, add(add(`if`(t=0 or irem(d, t)=0, d-d*t, d), d=divisors(j)) *A(n-j, t), j=1..n)/n) end: seq(A(n,10), n=0..50);

A[n_, t_] := A[n, t] = Module[{d, j}, If[n == 0, 1, Sum[Sum[If[t == 0 || Mod[d, t] == 0, d - d t, d], {d, Divisors[j]}] A[n - j, t], {j, 1, n}]/n]];
Table[A[n, 10], {n, 0, 50}] (* Jean-François Alcover, Dec 06 2020, after Alois P. Heinz *)

G.f.: Product_{i>=1} (1-x^(10*i))^10/(1-x^i).

Euler transform of period 10 sequence [1,1,1,1,1,1,1,1,1,-9, .. ].

A246650 Expansion of phi(x) * chi(-x) * psi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Aug 31 2014

sign

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

			G.f. = 1 + x - 2*x^2 + 2*x^4 - 3*x^5 - 2*x^6 + x^8 + 2*x^9 - 2*x^10 + ...
G.f. = q + q^4 - 2*q^7 + 2*q^13 - 3*q^16 - 2*q^19 + q^25 + 2*q^28 - ...

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, A121361, A122861, A123884, A129451, A131961, A131962, A131963, A131964.

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

Expansion of q^(-1/3) * eta(q^2)^4 * eta(q^6)^2 / (eta(q) * eta(q^3) * eta(q^4)^2) in powers of q.
a(2*n) = A129451(n). a(4*n) = A123884(n). a(4*n + 1) = A122861(n). a(4*n + 2) = -2 * A121361(n). a(4*n + 3) = 0.
a(8*n) = A131961(n). a(8*n + 1) = A097195(n). a(8*n + 2) = -2 * A131962(n). a(8*n + 4) = 2 * A131963(n). a(8*n + 6) = -2 * A131964(n).
a(16*n + 1) = A123884(n). a(16*n + 5) = -3 * A033687(n). a(16*n + 9) = 2 * A121361(n). a(16*n + 13) = 0.

A253625 Expansion of psi(q^2) * f(-q, q^2)^2 / f(-q, -q^5) in powers of q where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -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

Offset: 0

Michael Somos, Jan 06 2015

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).

			G.f. = 1 - 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 = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A033687, A033762, A093766, A097195, A107760, A112604, A112605, A121361, A121373, A122861, A123884, A253623, A253626.

A := Basis( ModularForms( Gamma1(12), 1), 81); A[1] - A[2] + 3*A[3] - A[4] + 3*A[5];

a[ n_] := If[ n < 1, Boole[n == 0], (-1)^ n Sum[(-1)^ Quotient[ d, 3] {1, 1, 0}[[ Mod[d, 3, 1] ]] {1, 2}[[ Mod[n/d, 2, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q^3] (QPochhammer[ -q^3, q^6] QPochhammer[ -q^2, q^2])^4, {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3]^2 EllipticTheta[ 2, 0, q]^2 / (EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, 0, q^(3/2)]), {q, 0, n}];

{a(n) = if( n<1, n==0, (-1)^n * sumdiv(n, d, (-1)^(d\3) * (d%3>0) * (2-(n\d)%2)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^4 * eta(x^6 + A)^8 / (eta(x^2 + A)^4 * eta(x^3 + A)^3 * eta(x^12 + A)^4), n))};

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

Expansion of psi(q^2)^2 * phi(q^3)^2 / (psi(q) * psi(q^3)) = f(-q) * f(-q^3) * (chi(q^3) / chi(-q^2))^4 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (-a(q) - 3*a(q^2) + 4*a(q^4)) / 6 = b(q^4) * (b(q) + 2*b(q^4)) / (3*b(q^2)) in powers of q where a(), b() are cubic AGM theta functions.
Expansion of eta(q) * eta(q^4)^4 * eta(q^6)^8 / (eta(q^2)^4 * eta(q^3)^3 * eta(q^12)^4) in powers of q.
Euler transform of period 12 sequence [ -1, 3, 2, -1, -1, -2, -1, -1, 2, 3, -1, -2, ...].
Moebius transform is period 12 sequence [ -1, 4, 0, 0, 1, 0, -1, 0, 0, -4, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 48^(-1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A253623.
a(n) = -b(n) where b() is multiplicative with b(2^e) = -3 if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f.: 1 + Sum_{k>0} (3 - (k mod 2)*4) * (x^k + x^(3*k)) / (1 + x^(2*k) + x^(4*k)).
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(3*k)) * (1 + x^(3*k))^4 / (1 - x^(2*k) + x^(4*k))^4.
a(n) = (-1)^n * A253626(n). a(2*n) = A107760(n). a(2*n + 1) = - A033762(n). a(3*n) = a(n). a(3*n + 1) = - A122861(n). a(4*n + 1) = - A112604(n). a(4*n + 2) = 3 * A033762(n). a(4*n + 3) = - A112605(n).
a(6*n + 1) = - A097195(n). a(6*n + 2) = 3 * A033687(n). a(6*n + 5) = 0. a(12*n + 1) = - A123884(n). a(12*n + 7) = -2 * A121361(n). a(12*n + 10) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Jun 08 2025

Typo in formula fixed by Colin Barker, Jan 08 2015

A363675 Numbers k such that the least common multiple of the degrees of the irreducible characters of S_k equals |S_k| = k!.

Original entry on oeis.org

0, 1, 6, 10, 21, 36, 66, 105, 120, 136, 190, 210, 276, 325, 465, 496, 561, 630, 666, 741, 780, 990, 1081, 1176, 1225, 1540, 1596, 1830, 2080, 2145, 2346, 2556, 2926, 3081, 3160, 3240, 3486, 3570, 3916, 4005, 4186, 4560, 4656, 4950, 5050, 5356, 5460, 5886, 6105

Offset: 1

Diego Martin Duro, Jun 14 2023

nonn

Intersection of the sequences of numbers k such that there exists a 2-core partition of k (A267137) and a 3-core partition of k (A000217).

Alois P. Heinz, Table of n, a(n) for n = 1..100
Diego Martin Duro, Arithmetic Properties of Character Degrees and the Generalised Knutson Index, arXiv:2306.12408 [math.RT], 2023.

Cf. A000142, A000217, A010054, A033687, A175595, A260682, A267137, A363676, A363701.

From Alois P. Heinz, Jun 16 2023: (Start)
{ k : A175595(k,2) > 0 and A175595(k,3) > 0 }.
{ k : A010054(k) > 0 and A033687(k) > 0 }. (End)

More terms from Alois P. Heinz, Jun 16 2023

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A113448 Expansion of (eta(q^2)^2 * eta(q^9) * eta(q^18)) / (eta(q) * eta(q^6)) in powers of q.

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A138806 Expansion of (theta_3(q) * theta_3(q^27) + theta_2(q) * theta_2(q^27) - 1) / 2 in powers of q.

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A244375 Expansion of (a(q) + 3*a(q^2) - 4*a(q^4)) / 6 in powers of q where a() is a cubic AGM theta function.

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A113063 Associated with theta series of hexagonal net with respect to a node.

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A123863 Expansion of (c(q^3) - c(q^6) - 2*c(q^12)) / 3 in powers of q where c(q) is a cubic AGM theta function.

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A112848 Expansion of eta(q)eta(q^2)eta(q^18)^2/(eta(q^6)*eta(q^9)) in powers of q.

A244375 Expansion of (a(q) + 3a(q^2) - 4a(q^4)) / 6 in powers of q where a() is a cubic AGM theta function.