cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A107503 Theta series of quadratic form with Gram matrix [ 8, 1, 1, -1; 1, 18, 5, 8; 1, 5, 18, 8; -1, 8, 8, 18].

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 0, 0, 0, 6, 4, 0, 10, 2, 6, 0, 8, 6, 0, 0, 0, 0, 8, 10, 0, 6, 6, 14, 0, 14, 20, 0, 0, 0, 0, 10, 24, 0, 16, 8, 28, 0, 28, 8, 0, 0, 0, 0, 34, 16, 0, 18, 14, 8, 0, 18, 40, 0, 0, 0, 0, 22, 26, 0, 40, 12, 44, 0, 38, 28, 0, 0, 0, 0, 36, 38, 0, 30
Offset: 0

Views

Author

N. J. A. Sloane, May 28 2005

Keywords

Comments

G.f. is theta_7 in the Parry 1979 reference on page 166. This theta series is an element of the space of modular forms on Gamma_0(169) of weight 2 and dimension 21. - Andy Huchala, May 13 2023

Examples

			G.f. = 1 + 2*q^8 + 6*q^18 + 4*q^20 + ...

Links

Crossrefs

Cf. A107498, A107499, A107500, A107501, A107502, A107504, A107505.

Programs

  • Magma
    prec := 90;
ls := [[8, 1, 1, -1], [1, 18, 5, 8], [1, 5, 18, 8], [-1, 8, 8, 18]];
S := Matrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
T := ThetaSeries(L, 48);
coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];
Coefficients(&+[coeffs[i]*B[i] :i in [1..13]]+&+[coeffs[i+1]*B[i] :i in [14..19]] + coeffs[22]*B[20] + coeffs[23]*B[21]); // Andy Huchala, May 13 2023

Extensions

Name clarified and more terms from Andy Huchala, May 13 2023

A107499 Theta series of quadratic form with Gram matrix [ 6, 2, 2, 1; 2, 18, 5, 9; 2, 5, 18, 9; 1, 9, 9, 24].

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 0, 0, 4, 4, 0, 8, 2, 10, 0, 10, 8, 0, 0, 0, 0, 8, 6, 0, 12, 6, 8, 0, 4, 22, 0, 0, 0, 0, 18, 32, 0, 10, 8, 22, 0, 26, 12, 0, 0, 0, 0, 36, 18, 0, 20, 14, 16, 0, 20, 34, 0, 0, 0, 0, 10, 22, 0, 42, 12, 42, 0, 44, 26, 0, 0, 0, 0, 38, 34, 0, 30
Offset: 0

Views

Author

N. J. A. Sloane, May 28 2005

Keywords

Comments

G.f. is theta_3 in the Parry 1979 reference on page 166. This theta series is an element of the space of modular forms on Gamma_0(169) of weight 2 and dimension 21. - Andy Huchala, May 13 2023

Examples

			G.f. = 1 + 2*q^6 + 4*q^18 + 4*q^20 + ...

Links

Crossrefs

Cf. A107498, A107500, A107501, A107502, A107503, A107504, A107505.

Programs

  • Magma
    prec := 90;
ls := [[6, 2, 2, 1], [2, 18, 5, 9], [2, 5, 18, 9], [1, 9, 9, 24]];
S := Matrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
T := ThetaSeries(L, 48);
coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];
Coefficients(&+[coeffs[i]*B[i] :i in [1..13]]+&+[coeffs[i+1]*B[i] :i in [14..19]] + coeffs[22]*B[20] + coeffs[23]*B[21]); // Andy Huchala, May 13 2023

Extensions

Name clarified and more terms from Andy Huchala, May 13 2023

A107500 Theta series of quadratic form with Gram matrix [ 10, 4, 4, 1; 4, 12, -1, 3; 4, -1, 12, 3; 1, 3, 3, 30].

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 4, 4, 2, 0, 0, 2, 0, 2, 0, 8, 0, 0, 14, 2, 8, 10, 0, 0, 18, 0, 6, 0, 18, 0, 0, 4, 24, 16, 22, 0, 0, 14, 0, 8, 0, 10, 0, 0, 20, 22, 22, 14, 0, 0, 30, 0, 14, 0, 30, 0, 0, 24, 22, 10, 48, 0, 0, 24, 0, 12, 0, 24, 0, 0, 46, 22, 60, 12, 0, 0, 32, 0
Offset: 0

Views

Author

N. J. A. Sloane, May 28 2005

Keywords

Comments

G.f. is theta_4 in the Parry 1979 reference on page 166. This theta series is an element of the space of modular forms on Gamma_0(169) of weight 2 and dimension 21. - Andy Huchala, May 13 2023

Examples

			G.f. = 1 + 2*q^10 + 4*q^12 + 4*q^14 + ...

Links

Crossrefs

Cf. A107498, A107499, A107501, A107502, A107503, A107504, A107505.

Programs

  • Magma
    prec := 90;
ls := [[10, 4, 4, 1], [4, 12, -1, 3], [4, -1, 12, 3], [1, 3, 3, 30]];
S := Matrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
T := ThetaSeries(L, 48);
coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];
Coefficients(&+[coeffs[i]*B[i] :i in [1..13]]+&+[coeffs[i+1]*B[i] :i in [14..19]] + coeffs[22]*B[20] + coeffs[23]*B[21]); // Andy Huchala, May 13 2023

Extensions

Name clarified and more terms from Andy Huchala, May 13 2023

A107502 Theta series of quadratic form with Gram matrix [ 4, 1, 0, -1; 1, 10, 0, 3; 0, 0, 26, 13; -1, 3, 13, 36].

Original entry on oeis.org

1, 0, 2, 0, 0, 2, 2, 0, 4, 0, 0, 2, 0, 2, 0, 6, 0, 0, 10, 8, 14, 12, 0, 0, 20, 0, 6, 0, 16, 0, 0, 8, 18, 18, 12, 0, 0, 12, 0, 8, 0, 6, 0, 0, 30, 22, 20, 10, 0, 0, 22, 0, 14, 0, 38, 0, 0, 22, 30, 18, 48, 0, 0, 30, 0, 12, 0, 22, 0, 0, 38, 16, 50, 30, 0, 0, 46, 0
Offset: 0

Views

Author

N. J. A. Sloane, May 28 2005

Keywords

Comments

G.f. is theta_6 in the Parry 1979 reference on page 166. This theta series is an element of the space of modular forms on Gamma_0(169) of weight 2 and dimension 21. - Andy Huchala, May 13 2023

Examples

			G.f. = 1 + 2*q^4 + 2*q^10 + 2*q^12 + ...

Links

Crossrefs

Cf. A107498, A107499, A107500, A107501, A107503, A107504, A107505.

Programs

  • Magma
    prec := 90;
ls := [[4, 1, 0, -1], [1, 10, 0, 3], [0, 0, 26, 13], [-1, 3, 13, 36]];
S := Matrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
T := ThetaSeries(L, 48);
coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];
Coefficients(&+[coeffs[i]*B[i] :i in [1..13]]+&+[coeffs[i+1]*B[i] :i in [14..19]] + coeffs[22]*B[20] + coeffs[23]*B[21]); // Andy Huchala, May 13 2023

Extensions

Name clarified and more terms from Andy Huchala, May 13 2023

A140782 a(n) = sigma(n) * Kronecker(13, n).

Original entry on oeis.org

1, -3, 4, 7, -6, -12, -8, -15, 13, 18, -12, 28, 0, 24, -24, 31, 18, -39, -20, -42, -32, 36, 24, -60, 31, 0, 40, -56, 30, 72, -32, -63, -48, -54, 48, 91, -38, 60, 0, 90, -42, 96, 44, -84, -78, -72, -48, 124, 57, -93, 72, 0, 54, -120, 72, 120, -80, -90, -60, -168, 62, 96, -104, 127, 0, 144, -68, 126, 96, -144
Offset: 1

Views

Author

Michael Somos, Jun 04 2008

Keywords

Comments

In the notation of Parry 1979 page 166, the g.f. is (theta_1 - theta_2) / 2 + theta_3 - theta_4 + theta_5 - theta_6 + theta_7 - theta_8 where theta_k is g.f. for A107497, ..., A107504.

Examples

			q - 3*q^2 + 4*q^3 + 7*q^4 - 6*q^5 - 12*q^6 - 8*q^7 - 15*q^8 + 13*q^9 + ...

Links

Crossrefs

Cf. A000203, A011583, A107497, A107498, A107499, A107500, A107501, A107502, A107503, A107504.

Programs

  • Mathematica
    Table[If[n==0, 0, DivisorSigma[1, n] JacobiSymbol[13, n]], {n, 100}] (* Indranil Ghosh, Jul 02 2017 *)
  • PARI
    {a(n) = if( n==0, 0, sigma(n) * kronecker( 13, n))}
  • PARI
    {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; (p^(e+1) - 1) / (p - 1) * kronecker( 13, p)^e)))}

Formula

a(n) is multiplicative with a(p^e) = (p^(e+1) - 1) / (p - 1) * Kronecker(13, p)^e.
G.f. is a period 1 Fourier series which satisfies f(-1 / (169 t)) = -169 (t/i)^2 f(t) where q = exp(2 Pi i t).
a(13*n) = 0. a(n) = A000203(n) * A011583(n). |a(n)| = A000203(n) unless 13 divides n.
a(n) = (A107497(n) - A107498(n)) / 2 + A107499(n) - A107500(n) + A107501(n) - A107502(n) + A107503(n) - A107504(n).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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