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 17 results. Next

A320067 Expansion of Product_{k>0} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 2, 6, 8, 10, 22, 26, 36, 60, 78, 106, 152, 202, 258, 370, 478, 602, 828, 1042, 1332, 1758, 2198, 2758, 3572, 4448, 5518, 7012, 8636, 10654, 13350, 16362, 19946, 24722, 30070, 36478, 44776, 54010, 65202, 79234, 95196, 114166, 137686, 164530, 196252, 235308, 279718, 332002
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2018

Keywords

Comments

Also the number of integer solutions (a_1, a_2, ..., a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n.

Links

Crossrefs

Cf. A000122, A029594, A033715, A320068, A320078, A320139, A320968, A320992.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[(&*[(1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2: j in [1..Floor(2*m/k)]]): k in [1..2*m]]))); // G. C. Greubel, Oct 29 2018
  • Mathematica
    nmax = 50; CoefficientList[Series[Product[EllipticTheta[3, 0, x^k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2018 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2, {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2018 *)
  • PARI
    m=50; x='x+O('x^m); Vec(1/(prod(k=1,2*m, prod(j=1,floor(2*m/k), (1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2 )))) \\ G. C. Greubel, Oct 29 2018

Formula

Expansion of Product_{k>0} eta(q^(2*k))^5 / (eta(q^k)*eta(q^(4*k)))^2.
a(n) ~ log(2)^(3/8) * exp(Pi*sqrt(n*log(2))) / (4 * Pi^(1/4) * n^(7/8)). - Vaclav Kotesovec, Oct 05 2018
Expansion of Product_{k>0} theta_4(q^(2*k))/theta_4(q^(2*k-1)), where theta_4() is the Jacobi theta function. - Seiichi Manyama, Oct 26 2018

A320234 Expansion of Product_{k=1..8} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 2, 6, 8, 10, 22, 26, 36, 58, 72, 96, 130, 164, 200, 268, 324, 376, 486, 552, 642, 796, 876, 992, 1198, 1294, 1436, 1682, 1794, 1964, 2268, 2428, 2556, 2980, 3116, 3304, 3876, 3940, 4252, 4896, 4996, 5348, 6164, 6260, 6668, 7686, 7808, 8120, 9378, 9490, 9762
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2018

Keywords

Comments

Also the number of integer solutions (a_1, a_2, ... , a_8) to the equation a_1^2 + 2*a_2^2 + ... + 8*a_8^2 = n.

Links

Crossrefs

Product_{k=1..m} theta_3(q^k): A000122 (m=1), A033715 (m=2), A029594 (m=3), A320139 (m=4), A320231 (m=5), A320232 (m=6), A320233 (m=7), this sequence (m=8), A320241 (m=9), A320242(m=10), A320246 (m=12), A320247 (m=16).
Cf. A320067, A320243.

A320231 Expansion of Product_{k=1..5} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 2, 6, 8, 10, 20, 20, 26, 38, 40, 48, 54, 60, 56, 80, 76, 60, 106, 76, 102, 132, 100, 128, 160, 174, 136, 210, 164, 164, 280, 160, 182, 256, 216, 232, 320, 204, 244, 408, 288, 288, 368, 316, 292, 518, 276, 264, 510, 310, 454, 480, 380, 408, 616, 524, 428, 656
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2018

Keywords

Comments

Also the number of integer solutions (a_1, a_2, ... , a_5) to the equation a_1^2 + 2*a_2^2 + ... + 5*a_5^2 = n.

Links

Crossrefs

Product_{k=1..m} theta_3(q^k): A000122 (m=1), A033715 (m=2), A029594 (m=3), A320139 (m=4), this sequence (m=5), A320232 (m=6), A320233 (m=7), A320234 (m=8).
Cf. A320067.

A320232 Expansion of Product_{k=1..6} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 2, 6, 8, 10, 22, 24, 30, 50, 56, 68, 94, 100, 108, 156, 156, 156, 214, 196, 214, 292, 252, 248, 374, 330, 344, 486, 380, 440, 640, 548, 506, 752, 624, 656, 988, 644, 720, 1080, 872, 872, 1220, 876, 984, 1598, 1052, 1096, 1566, 1290, 1310, 1936, 1260, 1264, 2198
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2018

Keywords

Comments

Also the number of integer solutions (a_1, a_2, ... , a_6) to the equation a_1^2 + 2*a_2^2 + ... + 6*a_6^2 = n.

Links

Crossrefs

Product_{k=1..m} theta_3(q^k): A000122 (m=1), A033715 (m=2), A029594 (m=3), A320139 (m=4), A320231 (m=5), this sequence (m=6), A320233 (m=7), A320234 (m=8).
Cf. A320067.

A320233 Expansion of Product_{k=1..7} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 2, 6, 8, 10, 22, 26, 34, 54, 68, 84, 114, 144, 156, 216, 256, 268, 350, 384, 414, 508, 564, 560, 686, 758, 736, 914, 966, 948, 1140, 1308, 1182, 1460, 1640, 1464, 1928, 2024, 1928, 2228, 2564, 2320, 2748, 3164, 2584, 3350, 3640, 3232, 3738, 4314, 3566, 4400
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2018

Keywords

Comments

Also the number of integer solutions (a_1, a_2, ... , a_7) to the equation a_1^2 + 2*a_2^2 + ... + 7*a_7^2 = n.

Links

Crossrefs

Product_{k=1..m} theta_3(q^k): A000122 (m=1), A033715 (m=2), A029594 (m=3), A320139 (m=4), A320231 (m=5), A320232 (m=6), this sequence (m=7), A320234 (m=8).
Cf. A320067.

A320138 Number of integer solutions to a^2 + 2*b^2 + 3*c^2 + 3*d^2 = n.

Original entry on oeis.org

1, 2, 2, 8, 10, 8, 24, 16, 10, 38, 8, 12, 48, 8, 32, 64, 26, 36, 70, 28, 24, 80, 28, 48, 96, 42, 40, 76, 48, 24, 112, 64, 58, 160, 68, 32, 126, 56, 44, 192, 56, 84, 176, 44, 60, 88, 80, 96, 208, 114, 74, 176, 72, 72, 172, 80, 112, 288, 88, 76, 224, 72, 112, 304, 90, 96
Offset: 0

Views

Author

Seiichi Manyama, Oct 06 2018

Keywords

Comments

a(n) > 0 for n >= 0.

Links

Crossrefs

Cf. A320139, A320140, A033712, A320188, A320189, A320190, A320191.

Formula

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^3)^2.

A320140 Number of integer solutions to a^2 + 2*b^2 + 3*c^2 + 5*d^2 = n.

Original entry on oeis.org

1, 2, 2, 6, 6, 6, 16, 8, 14, 26, 8, 32, 26, 8, 40, 16, 22, 40, 22, 32, 46, 40, 24, 48, 40, 42, 72, 50, 32, 64, 56, 28, 74, 48, 60, 112, 78, 24, 72, 76, 40, 144, 48, 48, 120, 50, 52, 48, 70, 98, 150, 128, 40, 84, 128, 52, 176, 120, 56, 208, 96, 72, 92, 72, 102, 192, 156
Offset: 0

Views

Author

Seiichi Manyama, Oct 06 2018

Keywords

Comments

a(n) > 0 for n >= 0.

Links

Crossrefs

Cf. A320138, A320139, A033712, A320188, A320189, A320190, A320191.

Formula

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^3) * theta_3(q^5).

A320188 Number of integer solutions to a^2 + 2*b^2 + 3*c^2 + 7*d^2 = n.

Original entry on oeis.org

1, 2, 2, 6, 6, 4, 12, 6, 6, 18, 12, 20, 26, 28, 20, 20, 34, 4, 30, 44, 20, 48, 44, 20, 20, 38, 16, 42, 62, 40, 56, 48, 34, 24, 72, 40, 70, 112, 36, 56, 68, 44, 40, 124, 60, 60, 124, 24, 66, 62, 54, 96, 92, 80, 64, 80, 64, 88, 136, 64, 76, 140, 52, 70, 166, 44, 104, 196, 44
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2018

Keywords

Comments

a(n) > 0 for n >= 0.

Links

Crossrefs

Cf. A320138, A320139, A320140, A033712, A320189, A320190, A320191.

Programs

  • Mathematica
    CoefficientList[Series[Product[EllipticTheta[3, 0, q^k], {k, 1, 3}]*EllipticTheta[3,0,q^7], {q, 0, 80}], q] (* G. C. Greubel, Oct 29 2018 *)
  • PARI
    q='q+O('q^80); Vec(prod(k=1,3, eta(q^(2*k))^5/(eta(q^k)* eta(q^(4*k)))^2 )*eta(q^(14))^5/(eta(q^7)* eta(q^(28)))^2 ) \\ G. C. Greubel, Oct 29 2018

Formula

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^3) * theta_3(q^7).

A320189 Number of integer solutions to a^2 + 2*b^2 + 3*c^2 + 8*d^2 = n.

Original entry on oeis.org

1, 2, 2, 6, 6, 4, 12, 4, 4, 18, 4, 20, 30, 12, 36, 24, 10, 32, 14, 24, 48, 32, 36, 40, 24, 18, 28, 34, 36, 60, 60, 28, 28, 40, 16, 56, 78, 44, 108, 68, 8, 72, 24, 40, 144, 60, 72, 112, 30, 46, 42, 64, 84, 116, 120, 40, 72, 84, 28, 116, 96, 60, 180, 68, 34, 120, 60, 64, 192, 80
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2018

Keywords

Comments

a(n) > 0 for n >= 0.

Links

Crossrefs

Cf. A320138, A320139, A320140, A033712, A320188, A320190, A320191.

Programs

  • Mathematica
    CoefficientList[Series[Product[EllipticTheta[3, 0, q^k], {k, 1, 3}]*EllipticTheta[3,0,q^8], {q, 0, 80}], q] (* G. C. Greubel, Oct 29 2018 *)
  • PARI
    q='q+O('q^80); Vec(prod(k=1,3, eta(q^(2*k))^5/(eta(q^k)* eta(q^(4*k)))^2 )*eta(q^(16))^5/(eta(q^8)* eta(q^(32)))^2 ) \\ G. C. Greubel, Oct 29 2018

Formula

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^3) * theta_3(q^8).

A320190 Number of integer solutions to a^2 + 2*b^2 + 3*c^2 + 9*d^2 = n.

Original entry on oeis.org

1, 2, 2, 6, 6, 4, 12, 4, 2, 16, 4, 12, 30, 16, 20, 40, 14, 8, 42, 8, 28, 60, 20, 32, 44, 22, 8, 46, 28, 36, 84, 36, 18, 40, 32, 8, 80, 40, 40, 120, 36, 24, 56, 40, 44, 140, 48, 48, 126, 22, 62, 64, 40, 68, 132, 72, 52, 120, 52, 60, 136, 56, 36, 160, 46, 64, 180, 40, 48
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2018

Keywords

Comments

a(n) > 0 for n >= 0.

Links

Crossrefs

Cf. A320138, A320139, A320140, A033712, A320188, A320189, A320191.

Programs

  • Mathematica
    CoefficientList[Series[Product[EllipticTheta[3, 0, q^k], {k, 1, 3}]*EllipticTheta[3,0,q^9], {q, 0, 80}], q] (* G. C. Greubel, Oct 29 2018 *)
  • PARI
    q='q+O('q^80); Vec(prod(k=1,3, eta(q^(2*k))^5/(eta(q^k)* eta(q^(4*k)))^2 )*eta(q^(18))^5/(eta(q^9)* eta(q^(36)))^2 ) \\ G. C. Greubel, Oct 29 2018

Formula

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^3) * theta_3(q^9).
Showing 1-10 of 17 results. Next

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

Content is available under The OEIS End-User License Agreement.