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.

Previous Showing 11-20 of 20 results.

A320247 Expansion of Product_{k=1..16} 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, 600, 822, 1032, 1310, 1720, 2140, 2656, 3418, 4222, 5172, 6510, 7922, 9636, 11928, 14424, 17268, 21088, 25236, 29996, 36222, 42824, 50544, 60252, 70830, 82832, 97732, 113956, 132242, 154866, 179164, 206396
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2018

Keywords

Comments

Also the number of integer solutions (a_1, a_2, ... , a_16) to the equation a_1^2 + 2*a_2^2 + ... + 16*a_16^2 = 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), A320246 (m=12), this sequence (m=16).
Cf. A320067.

A320967 Expansion of Product_{k>0} theta_3(q^k)/theta_4(q^k), where theta_3() and theta_4() are the Jacobi theta functions.

Original entry on oeis.org

1, 4, 12, 36, 92, 220, 508, 1108, 2332, 4776, 9492, 18420, 35036, 65324, 119708, 216044, 384204, 674236, 1168968, 2003460, 3397300, 5704148, 9487740, 15642676, 25577900, 41495032, 66817812, 106837112, 169677372, 267755836, 419948980, 654799316, 1015276412, 1565765892
Offset: 0

Views

Author

Seiichi Manyama, Oct 25 2018

Keywords

Links

Crossrefs

Self-convolution of A320968.
Cf. A000122, A002448, A007096, A301554, A320067, A320970.

Programs

  • Mathematica
    With[{nmax=50}, CoefficientList[Series[Product[EllipticTheta[3, 0, q^k]/EllipticTheta[4, 0, q^k], {k, 1, nmax+2}], {q, 0, nmax}], q]] (* G. C. Greubel, Oct 29 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(k=1,m+2, eta(q^(2*k))^6/(eta(q^k)^4* eta(q^(4*k))^2) )) \\ G. C. Greubel, Oct 29 2018

Formula

Expansion of Product_{k>0} eta(q^(2*k))^6 / (eta(q^k)^4*eta(q^(4*k))^2).

A320931 a(n) = [x^(n*(n+1)/2)] Product_{k=1..n} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 4, 12, 24, 80, 292, 966, 3876, 15554, 61608, 254612, 1065676, 4471672, 19074968, 82043172, 354365492, 1543432514, 6760146292, 29732837780, 131440491584, 583419967664, 2598585783488, 11615321544700, 52079369904384, 234157152231726, 1055628140278948, 4770576024205060
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 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*(n+1)/2.

Examples

			Solutions (a_1, a_2, ... , a_4) to the equation a_1^2 + 2*a_2^2 + ... + 4*a_4^2 = 10.
-------------------------------------------------------------------------------------
( 1,  1,  1,  1), ( 1,  1,  1, -1),
( 1,  1, -1,  1), ( 1,  1, -1, -1),
( 1, -1,  1,  1), ( 1, -1,  1, -1),
( 1, -1, -1,  1), ( 1, -1, -1, -1),
(-1,  1,  1,  1), (-1,  1,  1, -1),
(-1,  1, -1,  1), (-1,  1, -1, -1),
(-1, -1,  1,  1), (-1, -1,  1, -1),
(-1, -1, -1,  1), (-1, -1, -1, -1),
( 2,  1,  0,  1), ( 2,  1,  0, -1),
( 2, -1,  0,  1), ( 2, -1,  0, -1),
(-2,  1,  0,  1), (-2,  1,  0, -1),
(-2, -1,  0,  1), (-2, -1,  0, -1).

Links

Crossrefs

Cf. A000122, A000217, A173519, A320067, A320932.

Programs

  • Mathematica
    nmax = 25; Table[SeriesCoefficient[Product[EllipticTheta[3, 0, x^k], {k, 1, n}], {x, 0, n*(n+1)/2}], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2018 *)

Formula

a(n) ~ c * d^n / n^(7/4), where d = 4.818071572655... and c = 0.5869031198... - Vaclav Kotesovec, Oct 29 2018

A308286 Expansion of Product_{i>=1, j>=1} theta_3(x^(i*j)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 4, 12, 20, 40, 84, 140, 252, 456, 752, 1260, 2128, 3392, 5436, 8760, 13582, 21092, 32744, 49620, 75104, 113448, 168508, 249620, 368840, 538412, 783480, 1136652, 1634000, 2341280, 3344680, 4743684, 6706120, 9452392, 13245800, 18504888, 25777520, 35735376
Offset: 0

Views

Author

Ilya Gutkovskiy, May 18 2019

Keywords

Links

Crossrefs

Cf. A000005, A000122, A006171, A007425, A300446, A301554, A305050, A308288, A320067, A320908, A321241.

Programs

  • Mathematica
    nmax = 37; CoefficientList[Series[Product[Product[EllipticTheta[3, 0, x^(i j)], {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[Product[EllipticTheta[3, 0, x^k]^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} theta_3(x^k)^tau(k), where tau = number of divisors (A000005).
G.f.: Product_{i>=1, j>=1} (Sum_{k=-oo..+oo} x^(i*j*k^2)).
G.f.: Product_{i>=1, j>=1, k>=1} (1 - x^(i*j*k))*(1 + x^(i*j*k))^3/(1 + x^(2*i*j*k))^2.
G.f.: Product_{k>=1} (1 - x^k)^tau_3(k)*(1 + x^k)^(3*tau_3(k))/(1 + x^(2*k))^(2*tau_3(k)), where tau_3 = A007425.

A321027 Expansion of Product_{k>0} theta_3(q^(2*k))/theta_3(q^(2*k-1)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, -2, 6, -14, 28, -58, 114, -210, 384, -684, 1178, -2010, 3372, -5538, 9006, -14466, 22906, -35954, 55884, -85946, 131176, -198622, 298274, -444958, 659368, -970544, 1420362, -2066876, 2990680, -4305598, 6168154, -8793554, 12480718, -17637250, 24818530, -34785622
Offset: 0

Views

Author

Seiichi Manyama, Oct 26 2018

Keywords

Links

Crossrefs

Cf. A000122, A320067, A320098, A321026.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[EllipticTheta[3, 0, x^(2*k)] / EllipticTheta[3, 0, x^(2*k-1)], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 26 2018 *)

A329264 a(n) is the number of solutions of the infinite Diophantine equation Sum_{j>0} j^r*(k_j)^2 = n with k_j integers and r = 2.

Original entry on oeis.org

1, 2, 0, 0, 4, 4, 0, 0, 4, 4, 4, 0, 0, 12, 8, 0, 6, 16, 4, 0, 16, 8, 8, 0, 8, 24, 20, 0, 0, 52, 24, 0, 12, 32, 28, 8, 24, 12, 48, 16, 24, 68, 48, 8, 16, 96, 32, 16, 8, 68, 96, 32, 40, 68, 128, 32, 80, 88, 76, 48, 32, 156, 104, 64, 8, 224, 192, 40, 88, 152, 208
Offset: 0

Views

Author

Stefano Spezia, Nov 09 2019

Keywords

Examples

			a(16) = 6 since there are 6 integer solutions to 1^2*k1^2 + 2^2*k2^2 + 3^2*k3^2 + 4^2*k4^2 + ... = 16:
k1 = +-4 and k_j = 0 for j > 1;
k1 = 0, k2 = +-2 and k_j = 0 for j > 2;
k1 = k2 = k3 = 0, k4 = +-1 and k_j = 0 for j > 4.

Links

Crossrefs

Cf. A000041, A000122, A320067 (r = 1), A320068, A320078, A320968, A320992, A329265 (r = 3), A329266 (r = 4).

Programs

  • Mathematica
    nmax=70; r=2; CoefficientList[Series[Product[Product[(1-(-1)^n*q^(n*j^r))/(1+(-1)^n*q^(n*j^r)),{n,1,nmax}],{j,1,nmax}],{q,0,nmax}],q]

Formula

a(n) = [q^n] Product_{j>0} Product_{n>0} (1 - (-1)^n*q^(n*j^r)) / (1 + (-1)^n*q^(n*j^r)) with r = 2 (see Proposition 1.1 in Zhou and Sun).

A329265 a(n) is the number of solutions of the infinite Diophantine equation Sum_{j>0} j^r*(k_j)^2 = n with k_j integers and r = 3.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 0, 2, 6, 0, 0, 4, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 4, 2, 0, 2, 4, 0, 0, 4, 2, 8, 0, 4, 18, 0, 0, 8, 0, 4, 0, 4, 12, 0, 0, 0, 4, 2, 0, 8, 4, 0, 0, 0, 0, 8, 0, 4, 16, 0, 0, 12, 4, 4, 0, 0, 16, 0, 0, 8, 10, 16, 0, 8, 16, 0, 0, 0, 4, 18, 0, 0, 16
Offset: 0

Views

Author

Stefano Spezia, Nov 09 2019

Keywords

Examples

			a(9) = 6 since there are 6 integer solutions to 1^3*k1^2 + 2^3*k2^2 + ... = 9:
k1 = +-3 and k_j = 0 for j > 1;
k1 = -1, k2 = +-1 and k_j = 0 for j > 2;
k1 = 1, k2 = +-1 and k_j = 0 for j > 2.

Links

Crossrefs

Cf. A000041, A000122, A320067 (r = 1), A320068, A320078, A320968, A320992, A329264 (r = 2), A329266 (r = 4).

Programs

  • Mathematica
    nmax=85; r=3; CoefficientList[Series[Product[Product[(1-(-1)^n*q^(n*j^r))/(1+(-1)^n*q^(n*j^r)),{n,1,nmax}],{j,1,nmax}],{q,0,nmax}],q]

Formula

a(n) = [q^n] Product_{j>0} Product_{n>0} (1 - (-1)^n*q^(n*j^r)) / (1 + (-1)^n*q^(n*j^r)) with r = 3 (see Proposition 1.1 in Zhou and Sun).

A329266 a(n) is the number of solutions of the infinite Diophantine equation Sum_{j>0} j^r*(k_j)^2 = n with k_j integers and r = 4.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 4, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 8, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 8, 4, 4, 0, 0, 4, 0, 0
Offset: 0

Views

Author

Stefano Spezia, Nov 09 2019

Keywords

Examples

			a(25) = 6 since there are 6 integer solutions to 1^4*k1^2 + 2^4*k2^2 + ... = 25:
k1 = +-5 and k_j = 0 for j > 1;
k1 = -3, k2 = +-1 and k_j = 0 for j > 2;
k1 = 3, k2 = +-1 and k_j = 0 for j > 2.

Links

Crossrefs

Cf. A000041, A000122, A320067 (r = 1), A320068, A320078, A320968, A320992, A329264 (r = 2), A329265 (r = 3).

Programs

  • Mathematica
    nmax=87;r=4;CoefficientList[Series[Product[Product[(1-(-1)^n*q^(n*j^r))/(1+(-1)^n*q^(n*j^r)),{n,1,nmax}],{j,1,nmax}],{q,0,nmax}],q]

Formula

a(n) = [q^n] Product_{j>0} Product_{n>0} (1 - (-1)^n*q^(n*j^r)) / (1 + (-1)^n*q^(n*j^r)) with r = 4 (see Proposition 1.1 in Zhou and Sun).

A320248 Expansion of Product_{k=1..24} 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, 4446, 5512, 7002, 8614, 10616, 13292, 16260, 19792, 24496, 29724, 35976, 44062, 52992, 63780, 77296, 92518, 110532, 132848, 158036, 187674, 224066, 264960
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2018

Keywords

Comments

Also the number of integer solutions (a_1, a_2, ... , a_24) to the equation a_1^2 + 2*a_2^2 + ... + 24*a_24^2 = n.
a(24045) = 45676735553670596752038069309732400 and a(24046) = 45676724028345437854371347712212432. So a(24045) > a(24046).

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), A320246 (m=12), A320247 (m=16), this sequence (m=24).
Cf. A320067.

A321179 a(n) = [x^(n^2)] Product_{k=1..n} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 2, 14, 44, 174, 988, 4314, 20780, 126320, 692328, 3836166, 23160914, 135752866, 803203484, 4902966108, 29745996950, 181712320506, 1124481497694, 6965802854354, 43360326335154, 271658784580760, 1706393926177980, 10757142052998054, 68081390206251952, 432001821971576352
Offset: 0

Views

Author

Seiichi Manyama, Oct 29 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^2.

Examples

			Solutions (a_1, a_2, a_3) to the equation a_1^2 + 2*a_2^2 + 3*a_3^2 = 9.
------------------------------------------------------------------------
( 1,  2,  0), ( 1, -2,  0),
(-1,  2,  0), (-1, -2,  0),
( 2,  1,  1), ( 2,  1, -1),
( 2, -1,  1), ( 2, -1, -1),
(-2,  1,  1), (-2,  1, -1),
(-2, -1,  1), (-2, -1, -1),
( 3,  0,  0), (-3,  0,  0).

Links

Crossrefs

Cf. A000122, A000290, A320067, A320931, A321139.

Programs

  • Mathematica
    nmax = 20; Table[SeriesCoefficient[Product[EllipticTheta[3, 0, x^k], {k, 1, n}], {x, 0, n^2}], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2018 *)
  • PARI
    {a(n) = polcoeff(prod(i=1, n, 1+2*sum(j=1, sqrtint(n^2\i), x^(i*j^2)+x*O(x^(n^2)))), n^2)}

Formula

a(n) ~ c * d^n / n^(7/4), where d = 6.8137220913147... and c = 0.178176349247... - Vaclav Kotesovec, Oct 30 2018
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