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.

A281245 Expansion of Product_{k>=1} (1 + x^(7*k-1)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 1, 5, 5, 1, 0, 0, 0, 2, 7, 5, 1, 0, 0, 0
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 18 2017

Keywords

Links

Crossrefs

Cf. A262928, A147599, A281243, A281244.
Cf. A261612, A169975, A280454.
Cf. A109708, A281455, A281456, A281457, A281458, A280457, A281459.

Programs

  • Mathematica
    nmax = 200; CoefficientList[Series[Product[(1 + x^(7*k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 7] == 6, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

Formula

a(n) ~ exp(sqrt(n/21)*Pi) / (2^(13/7)*21^(1/4)*n^(3/4)) * (1 + (13*Pi/(336*sqrt(21)) - 3*sqrt(21)/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Jan 18 2017, extended Jan 24 2017
G.f.: Sum_{k>=0} x^(k*(7*k + 5)/2) / Product_{j=1..k} (1 - x^(7*j)). - Ilya Gutkovskiy, Nov 24 2020

A280457 Expansion of Product_{k>=0} (1 + x^(7*k+1)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 1, 4, 5, 2, 0, 0, 0, 1, 5, 7, 3, 0, 0, 0, 1, 5, 8, 5, 1, 0, 0, 1, 6, 10, 6, 1, 0, 0, 1, 6, 12, 9, 2, 0, 0, 1, 7, 14, 11, 3, 0, 0, 1, 7, 16, 15, 5
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 03 2017

Keywords

Comments

Number of partitions of n into distinct parts congruent to 1 mod 7.

Examples

			a(37) = 3 because we have [36, 1], [29, 8] and [22, 15].

Links

Crossrefs

Cf. A000700, A016993, A169975, A261612, A280454.
Cf. A262928, A147599, A281243, A281244.
Cf. A109703, A281245, A281455, A281456, A281457, A281458, A281459.

Programs

  • Mathematica
    nmax = 105; CoefficientList[Series[Product[(1 + x^(7 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 7] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

Formula

G.f.: Product_{k>=0} (1 + x^(7*k+1)).
a(n) ~ exp(Pi*sqrt(n)/sqrt(21))/(2*2^(1/7)*21^(1/4)*n^(3/4)) * (1 + (13*Pi/(336*sqrt(21)) - 3*sqrt(21)/(8*Pi)) / sqrt(n)). - Ilya Gutkovskiy, Jan 03 2017, extended by Vaclav Kotesovec, Jan 24 2017

A281455 Expansion of Product_{k>=1} (1 + x^(7*k-2)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 3, 0, 1, 0, 0, 2, 0, 3, 0, 1, 0, 0, 3, 0, 4, 0, 1, 1, 0, 4, 0, 4, 0, 1, 1, 0, 5, 0, 5, 0, 1, 2, 0, 7, 0, 5, 0, 1, 3, 0, 8, 0
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 22 2017

Keywords

Links

Crossrefs

Cf. A109707, A281245, A281456, A281457, A281458, A280457.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + x^(7*k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 7] == 5, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

Formula

a(n) ~ exp(sqrt(n/21)*Pi) / (2^(12/7)*21^(1/4)*n^(3/4)) * (1 - (3*sqrt(21)/(8*Pi) + 11*Pi/(336*sqrt(21))) / sqrt(n)). - Vaclav Kotesovec, Jan 22 2017, extended Jan 24 2017

A281457 Expansion of Product_{k>=1} (1 + x^(7*k-4)).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 0, 3, 0, 0, 2, 1, 0, 0, 3, 0, 0, 3, 1, 0, 1, 4, 0, 0, 4, 1, 0, 1, 4, 0, 0, 5, 1, 0, 2, 5, 0, 0, 7, 1, 0, 3, 5, 0, 0, 8, 1, 0, 5, 6, 0, 1
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 22 2017

Keywords

Links

Crossrefs

Cf. A109705, A281245, A281455, A281456, A281458, A280457.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + x^(7*k - 4)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 7] == 3, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

Formula

a(n) ~ exp(sqrt(n/21)*Pi) / (2^(10/7)*21^(1/4)*n^(3/4)) * (1 - (3*sqrt(21)/(8*Pi) + 23*Pi/(336*sqrt(21))) / sqrt(n)). - Vaclav Kotesovec, Jan 22 2017, extended Jan 24 2017

A281458 Expansion of Product_{k>=1} (1 + x^(7*k-5)).

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 3, 0, 2, 0, 0, 1, 0, 3, 0, 3, 0, 1, 1, 0, 4, 0, 4, 0, 1, 1, 0, 4, 0, 5, 0, 2, 1, 0, 5, 0, 7, 0, 3, 1, 0, 5, 0, 8, 0, 5, 1, 1, 6, 0, 10, 0, 6
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 22 2017

Keywords

Links

Crossrefs

Cf. A109704, A281245, A281455, A281456, A281457, A280457.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + x^(7*k - 5)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 7] == 2, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

Formula

a(n) ~ exp(sqrt(n/21)*Pi) / (2^(9/7)*21^(1/4)*n^(3/4)) * (1 - (3*sqrt(21)/(8*Pi) + 11*Pi/(336*sqrt(21))) / sqrt(n)). - Vaclav Kotesovec, Jan 22 2017, extended Jan 24 2017

A284502 Expansion of Product_{k>=0} (1 - x^(7*k+4)) in powers of x.

Original entry on oeis.org

1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 2, 0, 0, -1, -1, 0, 0, 2, 0, 0, -1, -1, 0, 0, 3, 0, 0, -1, -2, 0, 0, 3, 0, 0, -1, -3, 0, 0, 4, 1, 0, -1, -4, 0, 0, 4, 1, 0, -1, -5, 0, 0, 5, 2, 0, -1, -7, 0, 0, 5, 3, 0, -1, -8
Offset: 0

Views

Author

Seiichi Manyama, Mar 28 2017

Keywords

Links

Crossrefs

Cf. Product_{k>=0} (1 - x^(7*k+m)): A284499 (m=1), A284500 (m=2), A284501 (m=3), this sequence (m=4), A284503 (m=5), A284504 (m=6).
Cf. A281456.

Programs

  • Mathematica
    CoefficientList[Series[Product[1 - x^(7k + 4), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 28 2017 *)
  • PARI
    Vec(prod(k=0, 100, 1 - x^(7*k + 4)) + O(x^101)) \\ Indranil Ghosh, Mar 28 2017

Formula

a(n) = -(1/n) * Sum_{k=1..n} A284445(k) * a(n-k), a(0) = 1.

A281461 Expansion of Product_{k>=1} (1 + x^(7*k-3))*(1 + x^(7*k-4)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 2, 1, 0, 2, 2, 0, 1, 3, 1, 0, 3, 3, 0, 2, 5, 2, 1, 5, 5, 1, 3, 7, 3, 1, 7, 7, 1, 5, 11, 5, 2, 11, 11, 2, 7, 15, 7, 3, 15, 15, 4, 11, 22, 11, 6, 22, 22, 6, 15, 30, 15, 8, 30, 30, 9, 22, 42, 22, 13, 42, 42, 14, 30, 56
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 22 2017

Keywords

Comments

Convolution of A281456 and A281457.

Links

Crossrefs

Cf. A035435, A281456, A281457.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + x^(7*k-3))*(1 + x^(7*k-4)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(sqrt(2*n/21)*Pi) / (2^(5/4)*21^(1/4)*n^(3/4)) * (1 - (3*sqrt(21/2)/(8*Pi) + 23*Pi/(84*sqrt(42))) / sqrt(n)). - Vaclav Kotesovec, Jan 22 2017, extended Jan 24 2017
Showing 1-7 of 7 results.

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

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