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 10 results.

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

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)))).

A320098 Expansion of Product_{k>0} 1/theta_3(q^(2*k-1)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, -2, 4, -10, 18, -34, 64, -110, 188, -320, 524, -846, 1358, -2130, 3308, -5102, 7750, -11674, 17468, -25862, 38022, -55558, 80532, -116034, 166284, -236784, 335416, -472868, 663146, -925762, 1286920, -1780962, 2454792, -3370806, 4610656, -6284090, 8535868, -11554834
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2018

Keywords

Links

Crossrefs

Cf. A000122, A320068, A320078.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[1/Product[EllipticTheta[3, 0, x^(2*k-1)], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2018 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*j*(2*k - 1)))^2/((1 - x^((2*k - 1)*j))*(1 + x^((2*k - 1)*j))^3), {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(1/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

Convolution inverse of A320078.
Expansion of Product_{k>0} (eta(q^(2*k-1))*eta(q^(4*(2*k-1))))^2 / eta(q^(2*(2*k-1)))^5.

A320970 Expansion of Product_{k>0} theta_4(q^k)/theta_3(q^k), where theta_3() and theta_4() are the Jacobi theta functions.

Original entry on oeis.org

1, -4, 4, -4, 20, -28, 20, -52, 84, -104, 156, -180, 308, -460, 468, -684, 1028, -1308, 1592, -2084, 2940, -3668, 4564, -5716, 7556, -9912, 11484, -14616, 19252, -23548, 28316, -35188, 44724, -54532, 65996, -79948, 99784, -122796, 143972, -175372, 216524, -259996, 308004, -371140
Offset: 0

Views

Author

Seiichi Manyama, Oct 25 2018

Keywords

Links

Crossrefs

Cf. A000122, A002448, A320068, A320908, A320967.

Programs

  • Mathematica
    With[{nmax=80}, CoefficientList[Series[Product[EllipticTheta[4, 0, q^k]/EllipticTheta[3, 0, q^k], {k, 1, nmax+2}], {q, 0, nmax}], q]] (* G. C. Greubel, Oct 29 2018 *)
  • PARI
    m=80; q='q+O('q^m); Vec(1/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^k)^4*eta(q^(4*k))^2) / eta(q^(2*k))^6.
a(n) ~ (-1)^n * exp(Pi*sqrt(log(2)*n)) * (log(2))^(1/4) / (4*n^(3/4)). - Vaclav Kotesovec, Oct 26 2018

A320069 Expansion of 1/(theta_3(q) * theta_3(q^2)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, -2, 2, -4, 10, -16, 20, -32, 58, -86, 112, -164, 260, -368, 480, -672, 986, -1348, 1750, -2372, 3312, -4416, 5684, -7520, 10148, -13266, 16912, -21960, 28896, -37168, 46944, -60032, 77466, -98312, 123076, -155392, 197422, -247696, 307540, -384096, 481776, -598500
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2018

Keywords

Links

Crossrefs

Cf. A000122, A033715, A320068.

Programs

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

Formula

Convolution inverse of A033715.
a(n) ~ (-1)^n * exp(Pi*sqrt(n)) / (8 * n^(5/4)). - Vaclav Kotesovec, Oct 05 2018

A320070 Expansion of 1/(theta_3(q) * theta_3(q^2) * theta_3(q^3)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, -2, 2, -6, 14, -20, 32, -60, 98, -150, 232, -360, 558, -828, 1196, -1776, 2614, -3700, 5238, -7480, 10516, -14592, 20180, -27832, 38216, -51970, 70184, -94842, 127612, -170140, 226164, -300324, 396754, -521520, 683484, -893432, 1164330, -1511188, 1954756, -2524188
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2018

Keywords

Links

Crossrefs

Cf. A000122, A029594, A320068.

Programs

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

Formula

Convolution inverse of A029594.
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (4*sqrt(6)*n^(3/2)). - Vaclav Kotesovec, Oct 05 2018

A321026 Expansion of Product_{k>0} theta_3(q^(2*k-1))/theta_3(q^(2*k)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, -2, -2, 8, 2, -14, -6, 24, 20, -54, -30, 104, 42, -170, -94, 302, 170, -524, -254, 852, 422, -1406, -706, 2256, 1144, -3550, -1796, 5628, 2710, -8670, -4254, 13190, 6650, -20118, -9842, 30200, 14618, -44874, -22014, 66376, 32590, -97350, -47398, 141788, 68668
Offset: 0

Views

Author

Seiichi Manyama, Oct 26 2018

Keywords

Links

Crossrefs

Cf. A000122, A320068, A320078, A321027.

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).
Showing 1-10 of 10 results.

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