A035175 -id:A035175 - cp's OEIS frontend

A122855 Expansion of (phi(q^3)phi(q^5) + phi(q)phi(q^15))/2 in powers of q where phi(q) is a Ramanujan theta function.

1, 1, 0, 1, 1, 1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 1, 3, 2, 0, 2, 1, 0, 0, 2, 2, 1, 0, 1, 0, 0, 0, 2, 4, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 3, 1, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 1, 2, 0, 0, 5, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 3, 1, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 4, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 0

Michael Somos, Sep 14 2006

Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k>=0} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

			1 + q + q^3 + q^4 + q^5 + 2*q^8 + q^9 + q^12 + q^15 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Alexander Berkovich and Hamza Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, The Ramanujan Journal, Vol. 20 (2009), pp. 375-408; arXiv preprint, arXiv:math/0611300 [math.NT], 2006-2007.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

A035175(n) = a(4n).

Cf. A000122, A000700, A010054, A121373.

a[0] = 1; a[n_] := DivisorSum[n, KroneckerSymbol[-15, #]*(-1)^Boole[Mod[#, 4] == 2]&]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 07 2015, adapted from PARI *)

{a(n)=if(n<1, n==0, sumdiv(n, d, kronecker(-15,d)*(-1)^(d%4==2)))}

{a(n)= local(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, e-1, if(p<7, 1, if(p%15==2^valuation(p%15,2), e+1, 1-e%2))))))}

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2*eta(x^6+A)*eta(x^10+A)*eta(x^30+A)^2/ (eta(x+A)*eta(x^4+A)*eta(x^15+A)*eta(x^60+A)), n))}

Expansion of (eta(q^2)^2*eta(q^6)eta(q^10)eta(q^30)^2)/ (eta(q)eta(q^4)eta(q^15)eta(q^60)) in powers of q.
a(n) is multiplicative with a(2^e) = |e-1|, a(3^e)=a(5^e)=1, a(p^e) = e+1 if p == 1, 2, 4, 8 (mod 15), a(p^e) = (1+(-1)^e)/2 if p == 7, 11, 13, 14 (mod 15).
Euler transform of period 60 sequence [ 1, -1, 1, 0, 1, -2, 1, 0, 1, -2, 1, -1, 1, -1, 2, 0, 1, -2, 1, -1, 1, -1, 1, -1, 1, -1, 1, 0, 1, -4, 1, 0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, 1, 0, 2, -1, 1, -1, 1, -2, 1, 0, 1, -2, 1, 0, 1, -1, 1, -2, ...].
Moebius transform is period 60 sequence [ 1, -1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, -1, -1, 0, 1, 1, 0, -1, 0, 0, -1, -1, 0, 0, -1, 0, -1, -1, 0, -1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, 1, -1, 0, ...].
a(15n+7) = a(15n+11) = a(15n+13) = a(15n+14) = 0.
a(3n) = a(5n) = a(n).
G.f.: 1 + Sum_{k>0} Kronecker(-15,k) x^k/(1-(-x)^k).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(15) = 0.811155... . - Amiram Eldar, Nov 24 2023

A316569 a(n) = Jacobi (or Kronecker) symbol (n, 15).

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 0, 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 0, 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 0, 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 0, 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 0

Offset: 0

Jianing Song, Aug 05 2018

Period 15: repeat [0, 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1].
Also a(n) = Kronecker(-15, n).
This sequence is one of the three non-principal real Dirichlet characters modulo 15. The other two are Jacobi or Kronecker symbols (n, 45) (or (45, n)) and (n, 75) (or (-75, n)).
Note that (Sum_{i=0..14} i*a(i))/(-15) = 2 gives the class number of the imaginary quadratic field Q(sqrt(-15)).

Eric Weisstein's World of Mathematics, Kronecker Symbol
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1,-1,0,1,-1).

Cf. A035175 (inverse Moebius transform).

Kronecker symbols: A063524 ((n, 0) or (0, n)), A000012 ((n, 1) or (1, n)), A091337 ((n, 2) or (2, n) or (n, 8) or (8, n)), A102283 ((n, 3) or (-3, n)), A000035 ((n, 4) or (4, n) or (n, 16) or (16, n)), A080891 ((n, 5) or (5, n)), A109017 ((n, 6) or (-6, n)), A175629 ((n, 7) or (-7, n)), A011655 ((n, 9) or (9, n)), A011582 ((n, 11) or (-11, n)), A134667 ((n, 12) or (-12, n)), A011583 ((n, 13) or (13, n)), this sequence ((n, 15) or (-15, n)).

[KroneckerSymbol(-15, n): n in [0..100]]; // Vincenzo Librandi, Aug 28 2018

Array[ JacobiSymbol[#, 15] &, 90, 0] (* Robert G. Wilson v, Aug 06 2018 *)
PadRight[{},100,{0,1,1,0,1,0,0,-1,1,0,0,-1,0,-1,-1}] (* Harvey P. Dale, Feb 20 2023 *)

PARI
```
a(n) = kronecker(n, 15)
```

a(n) = 1 for n == 1, 2, 4, 8 (mod 15); -1 for n == 7, 11, 13, 14 (mod 15); 0 for n that are not coprime with 15.
Completely multiplicative with a(p) = a(p mod 15) for primes p.
a(n) = A102283(n)*A080891(n).
a(n) = a(n+15) = -a(-n) for all n in Z.
From Chai Wah Wu, Feb 16 2021: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) - a(n-5) + a(n-7) - a(n-8) for n > 7.
G.f.: (x^7 - x^5 + 2*x^4 - x^3 + x)/(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1). (End)

A123864 Expansion of (eta(q^3) * eta(q^5))^2 / (eta(q) * eta(q^15)) in powers of q.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 0, 4, 1, 2, 0, 3, 0, 0, 1, 5, 2, 2, 2, 3, 0, 0, 2, 4, 1, 0, 1, 0, 0, 2, 2, 6, 0, 4, 0, 3, 0, 4, 0, 4, 0, 0, 0, 0, 1, 4, 2, 5, 1, 2, 2, 0, 2, 2, 0, 0, 2, 0, 0, 3, 2, 4, 0, 7, 0, 0, 0, 6, 2, 0, 0, 4, 0, 0, 1, 6, 0, 0, 2, 5, 1, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 6, 2, 4, 2, 6, 0, 2, 0, 3, 0, 4, 0, 0

Offset: 0

Michael Somos, Oct 14 2006

Number 31 of the 74 eta-quotients listed in Table I of Martin (1996).

Multiplicative because this sequence is the inverse Moebius transform of a multiplicative sequence Kronecker(-15, n). - Andrew Howroyd, Jul 27 2018

			G.f. = 1 + q + 2*q^2 + q^3 + 3*q^4 + q^5 + 2*q^6 + 4*q^8 + q^9 + 2*q^10 + ...

Andrew Howroyd, Table of n, a(n) for n = 0..1000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 37.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.

Cf. A035175, A106406.

A := Basis( ModularForms( Gamma1(15), 1), 106); A[1] + A[2] + 2*A[3] + A[4] + 3*A[5] + A[6] + 2*A[7]; /* Michael Somos, Feb 10 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3] QPochhammer[ q^5])^2 / ( QPochhammer[ q] QPochhammer[ q^15]), {q, 0, n}]; (* Michael Somos, Feb 10 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], Sum[ KroneckerSymbol[ -15, d], { d, Divisors[ n]}]]; (* Michael Somos, Feb 10 2015 *)

{a(n) = if( n<1, n==0, sumdiv( n, d, kronecker( -15, d)))};

{a(n) = if( n<1, n==0, (qfrep( [2, 1; 1, 8],n, 1) + qfrep( [4, 1; 1, 4], n, 1))[n])};

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^5 + A))^2 / (eta(x + A) * eta(x^15 + A)), n))};

Euler transform of period 15 sequence [ 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -2, ...].
Moebius transform is period 15 sequence [ 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 0, ...].
G.f. A(q) satisfies 0 = f(A(q), A(q^2), A(q^4)) where f(u, v, w) = - v^3 + 4*u*v*w - 2*u*w^2 - u^2*w.
G.f.: Product_{k>0} ((1 - x^(3*k)) * (1 - x^(5*k)))^2 / ((1 - x^k) * (1 - x^(15*k))).
G.f.: (1/2) * (Sum_{n,m in Z} x^(n^2 + n*m + 4*m^2) + x^(2*n^2 + n*m + 2*m^2)).
a(15*n + 7) = a(15*n + 11) = a(15*n + 13) = a(15*n + 14) = 0. a(3*n) = a(n).
a(n) = A035175(n) unless n=0. a(n) = |A106406(n)| unless n=0.
G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = 15^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Feb 10 2015
a(n) = Sum_{d | n}  Kronecker(-15, d). - Andrew Howroyd, Jul 27 2018
From Amiram Eldar, Feb 20 2024: (Start)
Multiplicative with a(p^e) = 1 if p = 3 or 5, e + 1 if Kronecker(-15, p) = 1, and 1 - (e mod 2) if Kronecker(-15, p) = -1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(15). (End)

A106406 Expansion of (eta(q) * eta(q^15))^2 / (eta(q^3) * eta(q^5)) in powers of q.

Original entry on oeis.org

1, -2, -1, 3, -1, 2, 0, -4, 1, 2, 0, -3, 0, 0, 1, 5, -2, -2, 2, -3, 0, 0, -2, 4, 1, 0, -1, 0, 0, -2, 2, -6, 0, 4, 0, 3, 0, -4, 0, 4, 0, 0, 0, 0, -1, 4, -2, -5, 1, -2, 2, 0, -2, 2, 0, 0, -2, 0, 0, 3, 2, -4, 0, 7, 0, 0, 0, -6, 2, 0, 0, -4, 0, 0, -1, 6, 0, 0, 2

Offset: 1

Michael Somos, May 02 2005

Number 30 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = q - 2*q^2 - q^3 + 3*q^4 - q^5 + 2*q^6 - 4*q^8 + q^9 + 2*q^10 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A028955, A028957, A035175, A123864.

A := Basis( ModularForms( Gamma1(15), 1), 80); A[2] - 2*A[3] - A[4] + 3*A[5] - A[6] + 2*A[7]; /* Michael Somos, May 18 2015 */

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^15])^2 / (QPochhammer[ q^3] QPochhammer[ q^5]), {q, 0, n}]; (* Michael Somos, May 18 2015 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ #, 3] KroneckerSymbol[ n/#, 5] &]]; (* Michael Somos, May 18 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^15 + A)^2 / (eta(x^3 + A) * eta(x^5 + A)), n))};

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( d, 3) * kronecker( n/d, 5)))};

{a(n) = my(A, p, e, x); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==3 || p==5, (-1)^e, (p%15) != 2^(x = valuation( p%15, 2)), (e+1)%2, (e+1) * (-1)^(x*e))))};

{a(n) = if( n<1, 0, (qfrep([2, 1;1, 8],n, 1) - qfrep([4, 1;1, 4], n, 1))[n])}; /* Michael Somos, Aug 25 2006 */

Euler transform of period 15 sequence [-2, -2, -1, -2, -1, -1, -2, -2, -1, -1, -2, -1, -2, -2, -2, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -v^3 + 4 * u*v*w + 2 * u*w^2 + u^2*w.
a(n) is multiplicative with a(3^e) = a(5^e) = (-1)^e, a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11, 13, 14 (mod 15), a(p^e) = e+1 if p == 1, 4 (mod 15), a(p^e) = (e+1) * (-1)^e if p == 2, 8 (mod 15). - Michael Somos, Oct 19 2005
G.f.: (1/2) * (Sum_{n,m in Z} x^(n^2 + n*m + 4*m^2) - x^(2*n^2 + n*m + 2 *m^2)). - Michael Somos, Aug 25 2006
G.f.: Sum_{k>0} Kronecker(k, 3) * x^k * (1 - x^k) * (1 - x^(2*k)) / (1 - x^(5*k)) = Sum_{k>0} Kronecker(k, 5) * x^k * (1 - x^k) / (1 - x^(3*k)).
G.f.: x * Product_{k>0} ((1 - x^k) * (1 - x^(15*k)))^2 / ((1 - x^(3*k)) * (1 - x^(5*k))).
a(15*n + 7) = a(15*n + 11) = a(15*n + 13) = a(15*n + 14) = 0. a(3*n) = a(5*n) = -a(n).
A035175(n) = |a(n)|. a(n)>0 iff n in A028957. a(n)<0 iff n in A028955.
G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = 15^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, May 18 2015

A260649 Expansion of (phi(q^3) * phi(q^5) + phi(q) * phi(q^15)) / 2 - 1 in powers of q where phi(q) is a Ramanujan theta function.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 1, 3, 2, 0, 2, 1, 0, 0, 2, 2, 1, 0, 1, 0, 0, 0, 2, 4, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 3, 1, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 1, 2, 0, 0, 5, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 3, 1, 0, 2, 0, 2, 0, 0

Offset: 1

Michael Somos, Nov 12 2015

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

			G.f. = x + x^3 + x^4 + x^5 + 2*x^8 + x^9 + x^12 + x^15 + 3*x^16 + 2*x^17 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A035175, A122855.

a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -15, #] If[ Mod[#, 4] == 2, -1, 1] &]];
a[ n_] := If[ n < 1, 0, Times@@ (Which[# == 1, 1, # == 2, #2 - 1, # < 6, 1, KroneckerSymbol[#, -15] == 1, #2 + 1, True, 1 - Mod[#2, 2]]& @@@ FactorInteger[n])];
a[ n_] := SeriesCoefficient[QPochhammer[ q^2]^2 QPochhammer[ q^6] QPochhammer[ q^10] QPochhammer[ q^30]^2 / (QPochhammer[ q] QPochhammer[ q^4] QPochhammer[ q^15] QPochhammer[ q^60]) - 1, {q, 0, n}];

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-15, d) * (-1)^(d%4==2) ))};

{a(n) = if( n<1, 0, qfrep( [1, 0; 0, 15], n)[n] + qfrep( [3, 0; 0, 5], n)[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, e-1, p==3 || p==5, 1, kronecker(p, -15) == 1, e+1, 1-e%2 )))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A) * eta(x^10 + A) * eta(x^30 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^15 + A) * eta(x^60 + A)) - 1, n))};

Expansion of (eta(q^2)^2 * eta(q^6) * eta(q^10) * eta(q^30)^2) / (eta(q) * eta(q^4) * eta(q^15) * eta(q^60)) - 1 in powers of q.
a(n) is multiplicative with a(2^e) = |e-1|, a(3^e) = a(5^e) = 1, a(p^e) = e+1 if p == 1, 2, 4, 8 (mod 15), a(p^e) = (1 + (-1)^e)/2 if p == 7, 11, 13, 14 (mod 15).
Moebius transform of a period 60 sequence.
G.f.: Sum_{k>0} Kronecker(-15, k) x^k / (1 - (-x)^k).
a(n) = A122855(n) unless n=0.
a(3*n) = a(5*n) = a(n). a(4*n) = A035175(n). a(4*n + 2) = 0.
a(15*n + 7) = a(15*n + 11) = a(15*n + 13) = a(15*n + 14) = 0.

cp's OEIS Frontend

A122855 Expansion of (phi(q^3)phi(q^5) + phi(q)phi(q^15))/2 in powers of q where phi(q) is a Ramanujan theta function.

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A316569 a(n) = Jacobi (or Kronecker) symbol (n, 15).

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A123864 Expansion of (eta(q^3) * eta(q^5))^2 / (eta(q) * eta(q^15)) in powers of q.

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A106406 Expansion of (eta(q) * eta(q^15))^2 / (eta(q^3) * eta(q^5)) in powers of q.

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A260649 Expansion of (phi(q^3) * phi(q^5) + phi(q) * phi(q^15)) / 2 - 1 in powers of q where phi(q) is a Ramanujan theta function.

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