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 1-7 of 7 results.

A266686 Expansion of Product_{k>=1} (1 + x^k - x^(3*k)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 4, 5, 4, 6, 8, 9, 10, 11, 14, 16, 18, 21, 25, 28, 31, 36, 41, 48, 52, 59, 69, 77, 85, 96, 109, 121, 133, 151, 172, 189, 208, 231, 260, 287, 316, 350, 390, 432, 471, 521, 578, 636, 695, 764, 842, 924, 1009, 1107, 1218, 1330, 1449, 1584
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 02 2016

Keywords

Links

Crossrefs

Cf. A264905, A266647, A266650, A275820.

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[1+x^k-x^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[2]] = 1; p[[4]] = -1; Do[Do[p[[j+1]] = p[[j+1]] + p[[j - k + 1]] - If[j < 3*k, 0, p[[j - 3*k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* Vaclav Kotesovec, May 10 2018 *)

Formula

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + exp(-x) - exp(-3*x)) dx = 0.59698046904738615106237970379036510874974380079287087827737... . - Vaclav Kotesovec, Jan 05 2016

A266647 Expansion of Product_{k>=1} (1 + x^k + x^(3*k)) / (1 - x^k).

Original entry on oeis.org

1, 2, 4, 9, 15, 27, 46, 75, 118, 187, 285, 429, 639, 935, 1354, 1945, 2758, 3878, 5417, 7493, 10300, 14070, 19087, 25741, 34542, 46081, 61185, 80869, 106391, 139368, 181867, 236357, 306060, 394939, 507860, 650946, 831792, 1059600, 1345920, 1704880, 2153682
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 02 2016

Keywords

Comments

Convolution of A264905 and A000041.

Links

Crossrefs

Cf. A100405, A264905, A266648, A266649, A266650.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1+x^k+x^(3*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ sqrt(6*c + Pi^2) * exp(sqrt((4*c + 2*Pi^2/3)*n)) / (12*Pi*n), where c = Integral_{0..infinity} log(1 + exp(-x) + exp(-3*x)) dx = 0.9953865985263189816963357718655148864441174218433250148867... . - Vaclav Kotesovec, Jan 05 2016

A275820 Expansion of Product_{k>=1} (1 + x^(2*k) + x^(3*k)).

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 3, 1, 3, 3, 3, 2, 7, 3, 8, 7, 10, 7, 16, 8, 17, 17, 21, 17, 35, 22, 37, 36, 46, 37, 69, 46, 74, 71, 91, 81, 128, 96, 144, 139, 173, 154, 236, 185, 263, 257, 314, 286, 417, 345, 470, 462, 557, 517, 719, 617, 815, 802, 960, 904, 1211, 1068
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 15 2016

Keywords

Links

Crossrefs

Cf. A000726, A263401, A264905, A266686, A275821.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[1+x^(2*k)+x^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[3]] = 1; p[[4]] = 1; Do[Do[p[[j+1]] = p[[j+1]] + If[j < 2*k, 0, p[[j - 2*k + 1]]] + If[j < 3*k, 0, p[[j - 3*k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* Vaclav Kotesovec, May 10 2018 *)

Formula

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + exp(-2*x) + exp(-3*x)) dx = 0.60248650631158778882474716370201988195290074160793967143564...

A376631 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^(2*j)).

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 2, 0, 3, 0, 2, 1, 3, 1, 3, 1, 2, 3, 2, 3, 2, 4, 1, 5, 2, 5, 2, 6, 1, 7, 2, 7, 3, 6, 4, 7, 5, 6, 7, 6, 7, 7, 9, 5, 11, 5, 12, 6, 14, 5, 15, 6, 16, 7, 17, 7, 18, 9, 18, 11, 19, 12, 20, 14, 19, 17, 19, 19, 20, 23, 18, 27, 18, 29, 20, 32, 19
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

Links

Crossrefs

Cf. A000009, A053258, A053261, A264905, A333179, A376580, A376632, A376660.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2)*Product[1+x^(2*j), {j, 1, k}], {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k))*x^k]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[2*nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} (x^j + x^(3*j)).
a(n) ~ c * A376660^sqrt(n) / sqrt(n), where c = 1/(2*sqrt(3 - 4*sinh(arcsinh(3^(3/2)/2) / 3) / sqrt(3))) = 0.39098976711379944962936707496887239986756106886318...
a(n) ~ A376580(n) * (A376660/A376621)^sqrt(n).

A275821 Expansion of Product_{k>=1} (1 + x^(2*k) - x^(3*k)).

Original entry on oeis.org

1, 0, 1, -1, 1, 0, 1, -1, 1, -1, 3, -2, 3, -3, 2, -1, 4, -3, 4, -4, 7, -7, 7, -7, 9, -6, 11, -10, 10, -11, 15, -14, 18, -19, 21, -17, 24, -22, 26, -29, 35, -34, 42, -43, 43, -39, 52, -52, 59, -59, 74, -72, 79, -87, 93, -87, 107, -108, 118, -126, 149, -146
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 15 2016

Keywords

Links

Crossrefs

Cf. A000726, A263401, A264905, A266686, A275820.

Programs

  • Mathematica
    nmax=100; CoefficientList[Series[Product[1+x^(2*k)-x^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]
RootReduce[QPochhammer[Root[-1 + # + #^3 &, 1], x] QPochhammer[Root[-1 + # + #^3 &, 2], x] QPochhammer[Root[-1 + # + #^3 &, 3], x] + O[x]^70][[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[3]] = 1; p[[4]] = -1; Do[Do[p[[j + 1]] = p[[j + 1]] + If[j < 2 k, 0, p[[j - 2 k + 1]]] - If[j < 3 k, 0, p[[j - 3 k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* Vaclav Kotesovec, May 06 2018 *)

Formula

a(n) ~ (-1)^n * c^(1/4) * exp(sqrt(c*n)) / (2^(3/2)*sqrt(Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + 2*exp(-x) + exp(-2*x) - exp(-3*x)) dx = 1.522848148277623680909526566...

A276519 Expansion of Product_{k>=1} 1/(1 - x^(2*k) - x^(3*k)).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 4, 9, 10, 17, 19, 34, 37, 61, 75, 112, 138, 209, 256, 376, 478, 675, 866, 1222, 1566, 2175, 2830, 3873, 5055, 6900, 9011, 12213, 16045, 21599, 28429, 38191, 50290, 67341, 88884, 118669, 156751, 209018, 276200, 367734, 486376, 646688
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 15 2016

Keywords

Links

Crossrefs

Cf. A264905, A266686, A275820, A275821, A276526.

Programs

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

Formula

a(n) ~ c * p / r^n, where r = A075778 = 1/A060006 = 0.7548776662466927600495... is the real root of the equation r^3 + r^2 - 1 = 0, p = Product_{n>1} 1/(1 - r^(2*n) - r^(3*n)) = 3.820450591662541853... and c = 0.41149558866264576338190038... is the real root of the equation -1 + 8*c - 23*c^2 + 23*c^3 = 0.

A276526 Expansion of Product_{k>=1} 1/(1 - x^(2*k) + x^(3*k)).

Original entry on oeis.org

1, 0, 1, -1, 2, -2, 3, -4, 7, -8, 11, -15, 22, -27, 37, -51, 70, -90, 121, -162, 220, -288, 381, -512, 688, -902, 1197, -1598, 2127, -2809, 3722, -4949, 6581, -8699, 11519, -15301, 20305, -26862, 35581, -47208, 62591, -82859, 109756, -145506, 192856, -255388
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 16 2016

Keywords

Links

Crossrefs

Cf. A264905, A266686, A275820, A275821, A276519.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[1/Product[1-x^(2*k)+x^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ c * p / r^n, where r = -A075778 = -0.7548776662466927600495... is the real root of the equation r^3 - r^2 + 1 = 0, p = Product_{n>1} 1/(1 - r^(2*n) + r^(3*n)) = 1.9844809074648434... and c = 0.41149558866264576338190038... is the real root of the equation -1 + 8*c - 23*c^2 + 23*c^3 = 0.
Showing 1-7 of 7 results.

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