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 1-10 of 20 results. Next

A056945 Jacobi form of weight 12 and index 1 associated to a (nonexistent) lattice vector of norm 2 for the Leech lattice.

Original entry on oeis.org

1, 0, 0, -4, 6, 0, 0, 32736, 131076, 0, 0, 3669012, 9172952, 0, 0, 95691552, 188239518, 0, 0, 1142929524, 1959705000, 0, 0, 8506686816, 13293227112, 0, 0, 45763087664, 67073100864, 0, 0, 195387947712, 272567759508, 0, 0, 698077783656, 938807478318, 0, 0, 2176654050912
Offset: 0

Views

Author

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 16 2000

Keywords

Comments

Let J(h)=E_8*E_{4,1}+(2h-60)*phi_{12,1} be the Jacobi form of weight 12 and index 1 associated with a norm 2 vector of a Niemeier lattice of Coxeter number h. Let J(h)=sum_{n,r} c(4n-r^2) q^n*z^r. So a(n)=c(4m-r^2) for h=0.
Let N(h,n) be the number of vectors of norm 2n for the lattice, then we have N(h,n)=c(4n)+2*sum_{1<=r<=sqrt(4n)}c(4n-r^2) if h is the Coxeter number of a Niemeier lattice. Note that N(0,n)=a(4n)-2*sum a(4n-r^2)=A008408(n), for the Leech lattice! Note also a(3)<0 and a(n) is nonnegative for n<=1000, except 3.

References

  • Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser, 1985.

Crossrefs

Cf. A003785, A008408, A056946, A055747.

Formula

E_8*E_{4, 1}-60*phi_{12, 1}. The E's are Eisenstein-Jacobi series and phi_{12, 1} is the unique normalized Jacobi cusp form of weight 12 and index 1.
a(n) = A055747(n) - 60*A003785(n). - Sean A. Irvine, May 18 2022

A055749 Jacobi form of weight 12 and index 1 for the Niemeier lattice of type D_24.

Original entry on oeis.org

1, 0, 0, 88, 926, 0, 0, 24640, 118932, 0, 0, 3786312, 9240664, 0, 0, 94951872, 187974558, 0, 0, 1145140744, 1960908360, 0, 0, 8505386304, 13288248072, 0, 0, 45751234016, 67083722816, 0, 0, 195423783552, 272571256980, 0, 0
Offset: 0

Views

Author

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 12 2000

Keywords

References

  • Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser, 1985.

Crossrefs

Cf. A022055, A055747, A003785.

Formula

E_8*E_{4, 1}+32*phi_12. phi_12=unique Jacobi cusp form of weight 12 index 1.
G.f.: b(z) + 32*c(z) where b(z) is the g.f. for A055747 and c(z) is the g.f. for A003785. - Sean A. Irvine, Apr 05 2022

A056948 Jacobi form of weight 12 and index 1 associated with a nonexistent Niemeier lattice of Coxeter number 1.

Original entry on oeis.org

1, 0, 0, -2, 26, 0, 0, 32560, 130812, 0, 0, 3671562, 9174424, 0, 0, 95675472, 188233758, 0, 0, 1142977594, 1959731160, 0, 0, 8506658544, 13293118872, 0, 0, 45762829976, 67073331776, 0, 0, 195388726752, 272567835540, 0, 0, 698077270836, 938806082418, 0, 0, 2176652437872
Offset: 0

Views

Author

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 17 2000

Keywords

Comments

a(3) < 0 (apparently the only negative term) so the lattice cannot exist.
a(4n) + 2*Sum_{r=1..floor(sqrt(4n))} a(4*n - r^2) = A056947(n).

References

  • Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser 1985

Crossrefs

Cf. A003785, A055747, A056947.

Formula

E_8 * E_{4, 1} + (2*h - 60) * E_{12, 1} where h = 1.
a(n) = A055747(n) - 58*A003785(n). - Sean A. Irvine, May 18 2022

A055750 Jacobi form of weight 12 and index 1 for the Niemeier lattice of type D_12^2.

Original entry on oeis.org

1, 0, 0, 40, 446, 0, 0, 28864, 125268, 0, 0, 3725112, 9205336, 0, 0, 95337792, 188112798, 0, 0, 1143987064, 1960280520, 0, 0, 8506064832, 13290845832, 0, 0, 45757418528, 67078180928, 0, 0, 195405086592, 272569432212, 0, 0
Offset: 0

Views

Author

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 12 2000

Keywords

Crossrefs

A008690, A055749, A003785.

Formula

E_8*E_{4, 1}-16*phi_12.
G.f.: b(z) - 16*c(z) where b(z) is the g.f. for A055747 and c(z) is the g.f. for A003785. Sean A. Irvine, Apr 05 2022

A055751 Jacobi form of weight 12 and index 1 for Niemeier lattice of type A_24.

Original entry on oeis.org

1, 0, 0, 46, 506, 0, 0, 28336, 124476, 0, 0, 3732762, 9209752, 0, 0, 95289552, 188095518, 0, 0, 1144131274, 1960359000, 0, 0, 8505980016, 13290521112, 0, 0, 45756645464, 67078873664, 0, 0, 195407423712, 272569660308, 0, 0
Offset: 0

Views

Author

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 12 2000

Keywords

References

  • Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser 1985.

Crossrefs

A008689, A055747, A003785.

Formula

E_8*E_{4, 1}-10*phi_12.
G.f.: b(z) - 10*c(z) where b(z) is the g.f. for A055747 and c(z) is the g.f. for A003785. - Sean A. Irvine, Apr 05 2022

A055752 Jacobi form of weight 12 and index 1 for Niemeier lattice of type A_17*E_7 or D_10*E_7^2.

Original entry on oeis.org

1, 0, 0, 32, 366, 0, 0, 29568, 126324, 0, 0, 3714912, 9199448, 0, 0, 95402112, 188135838, 0, 0, 1143794784, 1960175880, 0, 0, 8506177920, 13291278792, 0, 0, 45758449280, 67077257280, 0, 0, 195401970432, 272569128084, 0, 0
Offset: 0

Views

Author

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 12 2000

Keywords

References

  • Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser, 1985.

Crossrefs

A008691, A055747, A003785.

Formula

E_8*E_{4, 1}-24*phi_12.
G.f.: b(z) - 24*c(z) where b(z) is the g.f. for A055747 and c(z) is the g.f. for A003785. - Sean A. Irvine, Apr 05 2022

A055753 Jacobi form of weight 12 and index 1 for Niemeier lattice of type A_15 D_9.

Original entry on oeis.org

1, 0, 0, 28, 326, 0, 0, 29920, 126852, 0, 0, 3709812, 9196504, 0, 0, 95434272, 188147358, 0, 0, 1143698644, 1960123560, 0, 0, 8506234464, 13291495272, 0, 0, 45758964656, 67076795456, 0, 0, 195400412352, 272568976020, 0, 0
Offset: 0

Views

Author

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 12 2000

Keywords

References

  • Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser,1985.

Crossrefs

A008692, A055747, A003785.

Formula

E_8*E_{4, 1}-28*phi_12.
G.f.: b(z) - 28*c(z) where b(z) is the g.f. for A055747 and c(z) is the g.f. for A003785. - Sean A. Irvine, Apr 05 2022

A055754 Jacobi form of weight 12 and index 1 for Niemeier lattice of type D_8^3.

Original entry on oeis.org

1, 0, 0, 24, 286, 0, 0, 30272, 127380, 0, 0, 3704712, 9193560, 0, 0, 95466432, 188158878, 0, 0, 1143602504, 1960071240, 0, 0, 8506291008, 13291711752, 0, 0, 45759480032, 67076333632, 0, 0, 195398854272, 272568823956, 0, 0
Offset: 0

Views

Author

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 12 2000

Keywords

References

  • Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser,1985.

Crossrefs

A008693, A055747, A003785.

Formula

E_8*E_{4, 1}-32*phi_12.
G.f.: b(z) - 32*c(z) where b(z) is the g.f. for A055747 and c(z) is the g.f. for A003785. - Sean A. Irvine, Apr 05 2022

A055756 Jacobi form of weight 12 and index 1 for Niemeier lattice of type A_12^2.

Original entry on oeis.org

1, 0, 0, 22, 266, 0, 0, 30448, 127644, 0, 0, 3702162, 9192088, 0, 0, 95482512, 188164638, 0, 0, 1143554434, 1960045080, 0, 0, 8506319280, 13291819992, 0, 0, 45759737720, 67076102720, 0, 0, 195398075232, 272568747924, 0, 0
Offset: 0

Views

Author

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 12 2000

Keywords

References

  • Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser,1985.

Crossrefs

A008694, A055747, A003785.

Formula

E_8*E_{4, 1}-34*phi_12.
G.f.: b(z) + 32*c(z) where b(z) is the g.f. for A055747 and c(z) is the g.f. for A003785. - Sean A. Irvine, Apr 05 2022

A055757 Jacobi form of weight 12 and index 1 for Niemeier lattice of type E_6^4 or A_11 D_7 E_6.

Original entry on oeis.org

1, 0, 0, 20, 246, 0, 0, 30624, 127908, 0, 0, 3699612, 9190616, 0, 0, 95498592, 188170398, 0, 0, 1143506364, 1960018920, 0, 0, 8506347552, 13291928232, 0, 0, 45759995408, 67075871808, 0, 0, 195397296192, 272568671892, 0, 0
Offset: 0

Views

Author

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 12 2000

Keywords

References

  • Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser,1985.

Crossrefs

A008695, A055747, A003785.

Formula

E_8*E_{4, 1}-36*phi_12.
G.f.: b(z) - 36*c(z) where b(z) is the g.f. for A055747 and c(z) is the g.f. for A003785. - Sean A. Irvine, Apr 05 2022
Showing 1-10 of 20 results. Next

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