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 1-3 of 3 results.

A035179 a(n) = Sum_{d|n} Kronecker(-11, d).

Original entry on oeis.org

1, 0, 2, 1, 2, 0, 0, 0, 3, 0, 1, 2, 0, 0, 4, 1, 0, 0, 0, 2, 0, 0, 2, 0, 3, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0, 1, 6, 0, 2, 2, 1, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 4, 0, 0, 0, 1, 0, 0, 2, 0, 4, 0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 2, 5, 0, 0, 0, 0, 0, 0
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

This is a member of an infinite family of odd weight level 11 multiplicative modular forms. g_1 = A035179, g_3 = A129522, g_5 = A065099, g_7 = A138661. - Michael Somos, Jun 07 2015
Half of the number of integer solutions to x^2 + x*y + 3*y^2 = n. - Michael Somos, Jun 05 2005
From Jianing Song, Sep 07 2018: (Start)
Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -11.
Inverse Moebius transform of A011582. (End)
Coefficients of Dedekind zeta function for the quadratic number field of discriminant -11. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Examples

			G.f. = x + 2*x^3 + x^4 + 2*x^5 + 3*x^9 + x^11 + 2*x^12 + 4*x^15 + x^16 + 2*x^20 + ...

References

  • Henry McKean and Victor Moll, Elliptic Curves, Cambridge University Press, 1997, page 202. MR1471703 (98g:14032).

Links

Crossrefs

Cf. A028609, A065099, A129522, A138661.
Moebius transform gives A011582.
Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(11), 1), 88); B := (-1 + A[1] + 2*A[2] + 4*A[4] + 2*A[5]) / 2; B; // Michael Somos, Jun 07 2015
  • Mathematica
    a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -11, #] &]]; (* Michael Somos, Jun 07 2015 *)
  • PARI
    {a(n) = if( n<1, 0, qfrep([2, 1; 1, 6], n, 1)[n])}; \\ Michael Somos, Jun 05 2005
  • PARI
    {a(n) = if( n<1, 0, direuler(p=2, n, 1 / ((1 - X) * (1 - kronecker( -11, p)*X))) [n])}; \\ Michael Somos, Jun 05 2005
  • PARI
    {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -11, d)))};

Formula

a(n) is multiplicative with a(11^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if p == 2, 6, 7, 8, 10 (mod 11), a(p^e) = e + 1 if p == 1, 3, 4, 5, 9 (mod 11). - Michael Somos, Jan 29 2007
Moebius transform is period 11 sequence [ 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, 0, ...]. - Michael Somos, Jan 29 2007
G.f.: Sum_{k>0} Kronecker(-11, k) * x^k / (1 - x^k). - Michael Somos, Jan 29 2007
A028609(n) = 2 * a(n) unless n = 0. - Michael Somos, Jun 24 2011
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(11) = 0.947225... . - Amiram Eldar, Oct 11 2022

A065099 Weight 5 level 11 cusp form with complex multiplication by Q(sqrt(11)) and trivial character.

Original entry on oeis.org

1, 0, 7, 16, -49, 0, 0, 0, -32, 0, 121, 112, 0, 0, -343, 256, 0, 0, 0, -784, 0, 0, 167, 0, 1776, 0, -791, 0, 0, 0, -553, 0, 847, 0, 0, -512, -2113, 0, 0, 0, 0, 0, 0, 1936, 1568, 0, -1918, 1792, 2401, 0, 0, 0, -718, 0, -5929, 0, 0, 0, 4487, -5488, 0, 0, 0, 4096
Offset: 1

Views

Author

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Nov 20 2001

Keywords

Comments

This is a member of an infinite family of odd weight level 11 multiplicative modular forms. g_1 = A035179, g_3 = A129522, g_5 = A065099, g_7 = A138661. - Michael Somos, Jun 07 2015

Examples

			G.f. = q + 7*q^3 + 16*q^4 - 49*q^5 - 32*q^9 + 121*q^11 + 112*q^12 - 343*q^15 + ...

Links

Crossrefs

Cf. A035179, A129522, A138661.

Programs

  • Magma
    A := Basis( CuspForms( Gamma1(11), 5), 71); A[1] + 7*A[3] + 16*A[4] - 49*A[5] - 32*A[9] + 121*A[11] + 112*A[12] - 343*A[15]; /* Michael Somos, Aug 26 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ With[{F1 = (QPochhammer[ q] QPochhammer[ q^11])^2, F2 = (QPochhammer[ q^2] QPochhammer[ q^22])^2, F3 = (QPochhammer[ q^2] QPochhammer[ q^22])^3, F4 = (QPochhammer[ q^4] QPochhammer[ q^44])^2}, (F1^4 + 8 q F1^3 F2 + 32 q^2 F1^2 F2^2 + 88 q^3 F1 F2^3 + 64 q^4 F2^4 + 96 q^6 F4 F2^3 + 128 q^5 F1 F4 (F2^2 + q^2 F2 F4 + q^4 F4^2)) / F3], {q, 0, n}]; (* Michael Somos, Jun 07 2015 *)
  • PARI
    { B(N,a,x,y,x2,y2)= a=vector(N); for (x=0,floor(sqrt(4*N)), for (y=0,floor(sqrt(4*N/11)),x2=x*x; y2=y*y; n=(x2+11*y2); if (n%4==0 && n<=4*N && n>0, w=(2*x2*x2-132*x2*y2+242*y2*y2)/32; a[n/4]+=w; if (x*y !=0, a[n/4]+=w)))); a }
  • PARI
    {a(n) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==11, 121^e, kronecker( -11, p)==-1, if( e%2, 0, p^(2*e)), for( x=1, sqrtint(4*p\11), if( issquare(4*p - 11*x^2, &y), break)); y = y^4 - 4 * p*y^2 + 2 * p^2; a0=1; a1=y; for( i=2, e, x=y*a1 - p^4*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Jun 08 2007 */
  • PARI
    {a(n) = my(A, F1, F2, F4); if( n<1, 0, n--; A = x * O(x^n); F1 = (eta(x + A) * eta(x^11 + A))^2; F2 = (eta(x^2 + A) * eta(x^22 + A))^2; F4 = (eta(x^4 + A) * eta(x^44 + A))^2; polcoeff( (F1^4 + 8 * x * F1^3*F2 + 32 * x^2 * F1^2*F2^2 + 88 * x^3 * F1*F2^3 + 64 * x^4 * F2^4 + 96 * x^6 * F4*F2^3 + 128 * x^5 * F1*F4 * (F2^2 + x^2 * F2*F4 + x^4 * F4^2)) / (eta(x^2 + A) * eta(x^22 + A))^3, n))}; /* Michael Somos, Jun 08 2007 */
  • PARI
    {a(n) = if( n<1, 0, n*=4; sum( y=0, sqrtint(n\11), if( issquare( n - 11 * y^2), if( (n > 11*y^2) && y, 2, 1) * (n^2 - 88 * n*y^2 + 968 * y^4) / 16)))}; /* Michael Somos, Jun 08 2007 */

Formula

G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^(5/2) (t/i)^5 f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 08 2007
a(n) is multiplicative with a(11^e) = 121^e, a(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 2, 6, 7, 8, 10 (mod 11), a(p^e) = a(p) * a(p^(e-1)) - p^4 * a(p^(e-2)) if p == 1, 3, 4, 5, 9 (mod 11) where a(p) = y^4 - 4 * p*y^2 + 2 * p^2 and 4*p = y^2 + 11 * x^2. - Michael Somos, Jun 08 2007

A129522 Expansion of unique weight 3 level 11 multiplicative cusp form in powers of q.

Original entry on oeis.org

1, 0, -5, 4, -1, 0, 0, 0, 16, 0, -11, -20, 0, 0, 5, 16, 0, 0, 0, -4, 0, 0, 35, 0, -24, 0, -35, 0, 0, 0, -37, 0, 55, 0, 0, 64, -25, 0, 0, 0, 0, 0, 0, -44, -16, 0, 50, -80, 49, 0, 0, 0, -70, 0, 11, 0, 0, 0, 107, 20, 0, 0, 0, 64, 0, 0, 35, 0, -175, 0, -133, 0, 0
Offset: 1

Views

Author

Michael Somos, Apr 19 2007, Jun 06 2007

Keywords

Comments

This is a member of an infinite family of odd weight level 11 multiplicative modular forms. g_1 = A035179, g_3 = A129522, g_5 = A065099, g_7 = A138661.

Examples

			G.f. = q - 5*q^3 + 4*q^4 - q^5 + 16*q^9 - 11*q^11 - 20*q^12 + 5*q^15 + 16*q^16 + ...

Crossrefs

Cf. A006571, A028609, A030200, A035179, A065099, A138661.

Programs

  • Magma
    A := Basis( CuspForms( Gamma1(11), 3), 73); A[1] - 5*A[3] + 4*A[4] - A[5]; /* Michael Somos, Mar 26 2015 */
  • Mathematica
    a[ n_] := Module[ {A, B}, B = QPochhammer[ q] QPochhammer[ q^11]; A = B / (q QPochhammer[ q^3] QPochhammer[ q^33]); SeriesCoefficient[ q B^3 (1 + 3 / A) Sqrt[ q (A + 1 + 3 / A)], {q, 0, n}]]; (* Michael Somos, Mar 26 2015 *)
  • PARI
    {a(n) = my(A, B); if( n<1, 0, n--; A = x * O(x^n); B = eta(x + A) * eta(x^11 + A); A = B /( x * eta(x^3 + A) * eta(x^33 + A)); A = B^3 * (1 + 3/A) * sqrt(x * (A + 1 + 3/A)); polcoeff(A, n))};
  • PARI
    {a(n) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==11, (-11)^e, kronecker( -11, p)==-1, if( e%2, 0, p^e), for( x=1, sqrtint(4*p\11), if( issquare(4*p - 11*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)))}; /* Michael Somos, Jun 06 2007 */
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); A = eta(x + A) * eta(x^11 + A); polcoeff( A^2 / subst(A + x * O(x^(n\2)), x, x^2) * (A^2 + 4*x * subst(A + x * O(x^(n\2)), x, x^2)^2 + 8 * x^3 * subst(A + x * O(x^(n\4)), x, x^4)^2), n))}; /* Michael Somos, Jun 06 2007 */

Formula

Expansion of (F(q)^2 + 4*F(q^2)^2 + 8*F(q^4)^2) * F(q)^2 / F(q^2) in powers of q where F(q) := eta(q) * eta(q^11) is the g.f. of A030200.
a(n) is multiplicative with a(11^e) = (-11)^e, a(p^e) = (1+(-1)^e)/2*p^e if p == 2, 6, 7, 8, 10 (mod 11), a(p^e) = a(p)*a(p^(e-1)) - p^2*a(p^(e-2)) if p == 1, 3, 4, 5, 9 (mod 11) where a(p) = y^2 - 2*p and 4*p = y^2 + 11*x^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
G.f.: (1/2) * Sum_{u, v in Z} (u*u - 3*v*v) * x^(u*u + u*v + 3*v*v). - Michael Somos, Jun 14 2007
Convolution of A006571 and A028609. - Michael Somos, Aug 14 2012
a(4*n + 2) = 0. - Michael Somos, Nov 11 2015
Showing 1-3 of 3 results.

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