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.

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A170961 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 8.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Aug 29 2010

Keywords

Comments

Product_{i=1..m} (1 + x^(4*i-1)) is the Poincaré polynomial for both Sp(2m) and O(2m+1).

References

  • H. Weyl, The Classical Groups, Princeton, 1946, see p. 238.

Links

Crossrefs

Cf. A000700, A147599, A169975, A170956-A170965.

Programs

  • Mathematica
    CoefficientList[Series[Product[1+x^(4i-1),{i,8}],{x,0,110}],x] (* Harvey P. Dale, Aug 22 2012 *)

Formula

a(n) = a(136-n). - Rick L. Shepherd, Mar 01 2013

A170962 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 9.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Aug 29 2010

Keywords

Comments

Product_{i=1..m} (1 + x^(4*i-1)) is the Poincaré polynomial for both Sp(2m) and O(2m+1).

References

  • H. Weyl, The Classical Groups, Princeton, 1946, see p. 238.

Links

Crossrefs

Cf. A000700, A147599, A169975, A170956-A170965.

Formula

a(n) = a(171-n). - Rick L. Shepherd, Mar 01 2013

A170963 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 10.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Aug 29 2010

Keywords

Comments

Product_{i=1..m} (1 + x^(4*i-1)) is the Poincaré polynomial for both Sp(2m) and O(2m+1).

References

  • H. Weyl, The Classical Groups, Princeton, 1946, see p. 238.

Links

Crossrefs

Cf. A000700, A147599, A169975, A170956-A170965.

Formula

a(n) = a(210-n). - Rick L. Shepherd, Mar 01 2013

A170964 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 11.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Aug 29 2010

Keywords

Comments

Product_{i=1..m} (1 + x^(4*i-1)) is the Poincaré polynomial for both Sp(2m) and O(2m+1).

References

  • H. Weyl, The Classical Groups, Princeton, 1946, see p. 238.

Links

Crossrefs

Cf. A000700, A147599, A169975, A170956-A170965.

Programs

  • Mathematica
    CoefficientList[Series[Product[1+x^(4k-1),{k,11}],{x,0,100}],x] (* Harvey P. Dale, Sep 19 2020 *)

Formula

a(n) = a(253-n). - 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

A170957 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 4.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Aug 29 2010

Keywords

Comments

Product_{i=1..m} (1 + x^(4*i-1)) is the Poincaré polynomial for both Sp(2m) and O(2m+1).

References

  • H. Weyl, The Classical Groups, Princeton, 1946, see p. 238.

Crossrefs

Cf. A000700, A147599, A169975, A170956-A170965.

A170958 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 5.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Aug 29 2010

Keywords

Comments

Product_{i=1..m} (1 + x^(4*i-1)) is the Poincaré polynomial for both Sp(2m) and O(2m+1).

References

  • H. Weyl, The Classical Groups, Princeton, 1946, see p. 238.

Crossrefs

Cf. A000700, A147599, A169975, A170956-A170965.

Programs

  • Mathematica
    CoefficientList[Series[Product[1+x^(4i-1),{i,5}],{x,0,60}],x] (* Harvey P. Dale, Mar 09 2019 *)
Previous Showing 21-30 of 35 results. Next

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

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