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

A320068 Expansion of Product_{k>0} 1/theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, -2, 2, -6, 12, -18, 30, -50, 84, -132, 198, -306, 476, -706, 1026, -1522, 2234, -3202, 4564, -6506, 9224, -12934, 17982, -24982, 34612, -47496, 64798, -88340, 119944, -161814, 217462, -291562, 389642, -518442, 687222, -908934, 1199040, -1575730, 2064466, -2699378, 3520540
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2018

Keywords

Links

Crossrefs

Cf. A000122, A004402, A320067, A320069, A320070, A320968, A320992.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[(&*[(1 + x^(2*k*j))^2/((1 - x^(k*j))*(1 + x^(k*j))^3): j in [1..(Floor(2*m/k)+1)]]): k in [1..2*m]]))); // G. C. Greubel, Oct 29 2018
  • Mathematica
    nmax = 50; CoefficientList[Series[1/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^(2*k*j))^2/((1 - x^(k*j))*(1 + x^(k*j))^3), {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2018 *)
  • PARI
    m=50; x='x+O('x^m); Vec(prod(k=1,2*m, prod(j=1,floor(2*m/k)+1, (1 + x^(2*k*j))^2/((1 - x^(k*j))*(1 + x^(k*j))^3) ))) \\ G. C. Greubel, Oct 29 2018

Formula

Convolution inverse of A320067.
Expansion of Product_{k>0} (eta(q^k)*eta(q^(4*k)))^2 / eta(q^(2*k))^5.
Expansion of Product_{k>0} theta_4(q^(2*k-1))/theta_4(q^(2*k)), where theta_4() is the Jacobi theta function. - Seiichi Manyama, Oct 26 2018

A320078 Expansion of Product_{k>0} theta_3(q^(2*k-1)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 0, 2, 6, 2, 4, 6, 8, 16, 8, 14, 26, 26, 24, 30, 58, 50, 60, 78, 90, 118, 104, 138, 192, 224, 204, 268, 366, 354, 412, 474, 596, 694, 724, 818, 1052, 1162, 1176, 1470, 1756, 1918, 2052, 2434, 2814, 3168, 3396, 3806, 4674, 5124, 5396, 6250, 7374, 7898, 8732
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2018

Keywords

Links

Crossrefs

Cf. A000122, A033716, A320067, A320098.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[EllipticTheta[3, 0, x^(2*k-1)], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2018 *)
nmax = 60; CoefficientList[Series[Product[(1 - x^((2*k-1)*j))*(1 + x^((2*k-1)*j))^3/(1 + x^(2*j*(2*k-1)))^2, {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2018 *)
  • PARI
    q='q+O('q^80); Vec(prod(k=1,50, eta(q^(2*(2*k-1)))^5/( eta(q^(2*k-1))* eta(q^(4*(2*k-1))))^2 ) ) \\ G. C. Greubel, Oct 29 2018

Formula

Expansion of Product_{k>0} eta(q^(2*(2*k-1)))^5 / (eta(q^(2*k-1))*eta(q^(4*(2*k-1))))^2.
a(n) ~ (log(2))^(1/4) * exp(Pi*sqrt(n*log(2)/2)) / (4*n^(3/4)). - Vaclav Kotesovec, Oct 07 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.

A320908 Expansion of Product_{k>=1} theta_4(x^k), where theta_4() is the Jacobi theta function.

Original entry on oeis.org

1, -2, -2, 2, 4, 6, -6, -2, -8, -12, 2, 6, 20, 14, 22, -2, -14, -34, -20, -42, -48, 34, 10, 50, 48, 80, 82, 52, -16, -30, -142, -130, -138, -226, -54, -70, 80, 190, 310, 238, 392, 178, 178, 86, -40, -148, -582, -506, -546, -680, -656, -126, -336, 262, 428, 930
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 23 2018

Keywords

Comments

Convolution of A288007 and A288098.
Convolution inverse of A301554.

Links

Crossrefs

Cf. A000005, A000203, A002448, A288007, A288098, A301554, A320067, A320068.

Programs

  • Magma
    m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(&*[(1-x^(j*k))/(1+x^(j*k)):j in [1..2*m]]): k in [1..2*m]]) )); // G. C. Greubel, Oct 29 2018
  • Maple
    with(numtheory): seq(coeff(series(mul(((1-x^k)/(1+x^k))^tau(k),k=1..n),x,n+1), x, n), n = 0 .. 60); # Muniru A Asiru, Oct 23 2018
  • Mathematica
    nmax = 55; CoefficientList[Series[Product[EllipticTheta[4, 0, x^k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 55; CoefficientList[Series[Product[((1 - x^k)/(1 + x^k))^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 55; CoefficientList[Series[Exp[-Sum[DivisorSigma[1, k] x^k (2 + x^k)/(k (1 - x^(2 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
  • PARI
    N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^k)/(1+x^k))^numdiv(k))) \\ Seiichi Manyama, Oct 25 2018

Formula

G.f.: Product_{i>=1, j>=1} (1 - x^(i*j))/(1 + x^(i*j)).
G.f.: Product_{k>=1} ((1 - x^k)/(1 + x^k))^d(k), where d(k) is the number of divisors of k (A000005).
G.f.: exp(-Sum_{k>=1} sigma(k)*x^k*(2 + x^k)/(k*(1 - x^(2*k)))).

A320241 Expansion of Product_{k=1..9} 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, 76, 100, 142, 180, 220, 312, 376, 448, 602, 696, 834, 1056, 1204, 1392, 1734, 1942, 2188, 2654, 2898, 3248, 3860, 4180, 4540, 5376, 5704, 6176, 7242, 7532, 8184, 9444, 9868, 10480, 12168, 12544, 13348, 15554, 15832, 16816, 19430
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2018

Keywords

Comments

Also the number of integer solutions (a_1, a_2, ... , a_9) to the equation a_1^2 + 2*a_2^2 + ... + 9*a_9^2 = 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), A320234 (m=8), this sequence (m=9), A320242 (m=10).
Cf. A320067.

A320242 Expansion of Product_{k=1..10} 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, 104, 146, 192, 236, 332, 420, 500, 674, 816, 986, 1256, 1488, 1752, 2174, 2566, 2940, 3550, 4102, 4640, 5528, 6292, 6948, 8160, 9172, 10060, 11618, 12840, 13980, 15940, 17590, 18844, 21252, 23308, 24772, 27926, 30360, 31932
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2018

Keywords

Comments

Also the number of integer solutions (a_1, a_2, ... , a_10) to the equation a_1^2 + 2*a_2^2 + ... + 10*a_10^2 = 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), A320234 (m=8), A320241 (m=9), this sequence (m=10).
Cf. A320067.

A320246 Expansion of Product_{k=1..12} 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, 200, 252, 360, 456, 564, 770, 940, 1178, 1532, 1852, 2256, 2858, 3430, 4100, 5086, 5982, 7076, 8612, 10040, 11672, 13960, 16068, 18496, 21866, 24796, 28288, 32924, 37074, 41876, 48156, 53732, 60014, 68546, 75836, 83996
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2018

Keywords

Comments

Also the number of integer solutions (a_1, a_2, ... , a_12) to the equation a_1^2 + 2*a_2^2 + ... + 12*a_12^2 = 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), A320234 (m=8), A320241 (m=9), A320242 (m=10), this sequence (m=12), A320247 (m=16).
Cf. A320067.
Showing 1-10 of 20 results. Next

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