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.

A169975 Expansion of Product_{i>=0} (1 + x^(4*i+1)).

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 3, 2, 0, 1, 3, 3, 1, 1, 4, 4, 1, 1, 4, 5, 2, 1, 5, 7, 3, 1, 5, 8, 5, 2, 6, 10, 6, 2, 6, 12, 9, 3, 7, 14, 11, 4, 7, 16, 15, 6, 8, 19, 18, 8, 9, 21, 23, 11, 10, 24, 27, 14, 11, 27, 34, 19, 13, 30, 39, 24, 15, 33, 47, 31, 18, 37, 54, 38
Offset: 0

Views

Author

N. J. A. Sloane, Aug 29 2010

Keywords

Comments

Number of partitions into distinct parts of the form 4*k+1.
In general, if a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} (1 + x^(a*k + b)), then a(n) ~ exp(Pi*sqrt(n/(3*a))) / (2^(1 + b/a) * (3*a)^(1/4) * n^(3/4)) [Meinardus, 1954]. - Vaclav Kotesovec, Aug 26 2015
Convolution of A147599 and A169975 is A000700. - Vaclav Kotesovec, Jan 18 2017

Links

Crossrefs

Cf. A000700, A261612, A280454, A280456, A280457.
Cf. A015128, A080054, A261610, A261611.
Cf. A000041, A000009, A035382, A035451.
Cf. A170966-A170975.
Cf. A262928, A147599, A281243, A281244, A281245.

Programs

  • Mathematica
    nmax = 200; CoefficientList[Series[Product[(1 + x^(4*k+1)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 26 2015 *)
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 4] == 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.: Sum_{n>=0} (x^(2*n^2 - n) / Product_{k=1..n} (1 - x^(4*k))). - Joerg Arndt, Mar 10 2011
G.f.: G(0)/x where G(k) = 1 - 1/(1 - 1/(1 - 1/(1+(x)^(4*k+1))/G(k+1) )); (recursively defined continued fraction, see A006950). - Sergei N. Gladkovskii, Jan 28 2013
a(n) ~ exp(Pi*sqrt(n)/(2*sqrt(3))) / (2^(7/4) * 3^(1/4) * n^(3/4)) * (1 - (3*sqrt(3)/(4*Pi) + Pi/(192*sqrt(3))) / sqrt(n)). - Vaclav Kotesovec, Aug 26 2015, extended Jan 18 2017

A170975 Expansion of Product_{i=0..m-1} (1 + x^(4*i+1)) for m = 12.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 3, 2, 0, 1, 3, 3, 1, 1, 4, 4, 1, 1, 4, 5, 2, 1, 5, 7, 3, 1, 5, 8, 5, 2, 6, 10, 6, 1, 5, 12, 9, 2, 5, 13, 11, 3, 4, 14, 15, 5, 4, 15, 17, 7, 4, 15, 21, 10, 4, 15, 23, 13, 4, 15, 27, 17, 5, 14, 28, 21, 6, 13, 31, 26, 8, 12, 31, 30, 11, 11
Offset: 0

Views

Author

N. J. A. Sloane, Aug 29 2010

Keywords

Links

Crossrefs

Cf. A169975, A170966-A170984.

Programs

  • Magma
    m:=12; R:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (&*[1+x^(4*j+1): j in [0..m-1]]) )); // G. C. Greubel, Feb 24 2019
  • Maple
    seq(coeff(mul((1+x^(4*i+1)),i=0..11),x,n),n=0..100); # Nathaniel Johnston, Jun 24 2011
  • Mathematica
    With[{m=12}, CoefficientList[Series[Product[(1 + x^(4*j+1)), {j,0,m-1}], {x,0,100}],x]] (* G. C. Greubel, Feb 24 2019 *)
  • PARI
    m=12; my(x='x+O('x^(100))); Vec(prod(j=0,m-1, 1+x^(4*j+1) )) \\ G. C. Greubel, Feb 24 2019
  • Sage
    m=12; ( prod(1+x^(4*j+1) for j in (0..m-1)) ).series(x, 100).coefficients(x, sparse=False) # G. C. Greubel, Feb 24 2019

Formula

a(n) = a(276-n). - Rick L. Shepherd, Mar 01 2013

Extensions

Typo in Maple program fixed and b-file extended 9 terms by Rick L. Shepherd, Mar 01 2013

A170971 Expansion of Product_{i=0..m-1} (1 + x^(4*i+1)) for m = 8.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Aug 29 2010

Keywords

Links

Crossrefs

Cf. A169975, A170966-A170984.

Formula

a(n) = a(120-n). - Rick L. Shepherd, Mar 01 2013

A170972 Expansion of Product_{i=0..m-1} (1 + x^(4*i+1)) for m = 9.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 3, 2, 0, 1, 3, 3, 1, 1, 4, 4, 1, 1, 4, 5, 2, 0, 4, 7, 3, 0, 3, 7, 5, 1, 3, 8, 6, 1, 2, 8, 8, 2, 2, 8, 9, 3, 1, 7, 11, 5, 1, 7, 11, 6, 1, 5, 12, 8, 1, 4, 11, 9, 2, 3, 11, 11, 3, 2, 9, 11, 4, 1, 8, 12, 5, 1, 6, 11, 7, 1, 5, 11, 7, 1, 3, 9, 8, 2, 2
Offset: 0

Views

Author

N. J. A. Sloane, Aug 29 2010

Keywords

Links

Crossrefs

Cf. A169975, A170966-A170984.

Programs

  • Mathematica
    CoefficientList[Series[Product[1+x^(4i+1),{i,0,8}],{x,0,100}],x] (* Harvey P. Dale, Jun 17 2013 *)

Formula

a(n) = a(153-n). - Rick L. Shepherd, Mar 01 2013

A170973 Expansion of Product_{i=0..m-1} (1 + x^(4*i+1)) for m = 10.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 3, 2, 0, 1, 3, 3, 1, 1, 4, 4, 1, 1, 4, 5, 2, 1, 5, 7, 3, 0, 4, 8, 5, 1, 4, 9, 6, 1, 3, 10, 9, 2, 3, 10, 10, 3, 2, 10, 13, 5, 2, 10, 14, 7, 2, 9, 16, 9, 2, 8, 16, 11, 2, 7, 18, 14, 3, 5, 16, 16, 5, 4, 16, 18, 6, 3, 14, 19, 9, 3, 13, 20, 10
Offset: 0

Views

Author

N. J. A. Sloane, Aug 29 2010

Keywords

Links

Crossrefs

Cf. A169975, A170966-A170984.

Programs

  • Mathematica
    CoefficientList[Series[Product[1+x^(4k+1),{k,0,9}],{x,0,100}],x] (* Harvey P. Dale, Aug 03 2021 *)

Formula

a(n) = a(190-n). - Rick L. Shepherd, Mar 01 2013

A170974 Expansion of Product_{i=0..m-1} (1 + x^(4*i+1)) for m = 11.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 3, 2, 0, 1, 3, 3, 1, 1, 4, 4, 1, 1, 4, 5, 2, 1, 5, 7, 3, 1, 5, 8, 5, 1, 5, 10, 6, 1, 4, 11, 9, 2, 4, 12, 11, 3, 3, 12, 14, 5, 3, 13, 16, 7, 3, 12, 19, 10, 3, 12, 20, 12, 3, 11, 23, 16, 4, 10, 23, 19, 5, 8, 24, 23, 7, 7, 23, 25, 10, 6, 23
Offset: 0

Views

Author

N. J. A. Sloane, Aug 29 2010

Keywords

Links

Crossrefs

Cf. A169975, A170966-A170984.

Programs

  • Mathematica
    CoefficientList[Series[Product[1+x^(4i+1),{i,0,10}],{x,0,100}],x] (* Harvey P. Dale, Apr 27 2025 *)

Formula

a(n) = a(231-n). - Rick L. Shepherd, Mar 01 2013

A170967 Expansion of Product_{i=0..m-1} (1 + x^(4*i+1)) for m = 4.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1
Offset: 0

Views

Author

N. J. A. Sloane, Aug 29 2010

Keywords

Crossrefs

Cf. A169975, A170966-A170984.

Programs

  • Mathematica
    CoefficientList[Series[Product[1+x^(4i+1),{i,0,3}],{x,0,30}],x] (* Harvey P. Dale, Jul 11 2019 *)

A170968 Expansion of Product_{i=0..m-1} (1 + x^(4*i+1)) for m = 5.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 0, 2, 2, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1
Offset: 0

Views

Author

N. J. A. Sloane, Aug 29 2010

Keywords

Crossrefs

Cf. A169975, A170966-A170984.

A170969 Expansion of Product_{i=0..m-1} (1 + x^(4*i+1)) for m = 6.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 3, 2, 0, 0, 2, 3, 1, 0, 2, 3, 1, 0, 1, 3, 2, 0, 1, 3, 2, 0, 0, 2, 3, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1
Offset: 0

Views

Author

N. J. A. Sloane, Aug 29 2010

Keywords

Crossrefs

Cf. A169975, A170966-A170984.

A170970 Expansion of Product_{i=0..m-1} (1 + x^(4*i+1)) for m = 7.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Aug 29 2010

Keywords

Crossrefs

Cf. A169975, A170966-A170984.

Programs

  • Mathematica
    CoefficientList[Series[Product[1+x^(4i+1),{i,0,6}],{x,0,100}],x] (* Harvey P. Dale, Jun 20 2015 *)
Showing 1-10 of 10 results.

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

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