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 1-6 of 6 results.

A059777 Number of self-conjugate three-quadrant Ferrers graphs that partition n.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 16, 19, 24, 31, 39, 47, 58, 72, 89, 107, 129, 158, 192, 228, 273, 329, 393, 465, 551, 655, 776, 911, 1070, 1261, 1480, 1726, 2014, 2354, 2742, 3180, 3688, 4279, 4954, 5716, 6590, 7603, 8754, 10049, 11532
Offset: 0

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Author

N. J. A. Sloane, Feb 21 2001

Keywords

References

  • G. E. Andrews, Three-quadrant Ferrers graphs, Indian J. Math., 42 (No. 1, 2000), 1-7.

Links

Crossrefs

Cf. A002129, A006950, A059776, A059778, A316675.

Programs

  • Maple
    mul((1+q^(2*n+3))/(1-q^(2*n+2)), n=0..101); # g.f.
  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k + 1))/(1 - x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 26 2016 *)

Formula

G.f.: 1/((1+x)*Sum_{k>=0} (-x)^(k*(k+1)/2)). [Corrected by N. J. A. Sloane, Jul 10 2022 at the suggestion of Eduardo Brietzke.] a(n) = (1/n)*Sum_{k=1..n} (-1)^(k+1)*(A002129(k)-1)*a(n-k). A006950(n) = a(n-1) + a(n), n > 0. - Vladeta Jovovic, Sep 22 2002
G.f.: 1/((1+x)*G(0)), where G(k)= 1 - x^(2*k+1)/(1 - x^(2*k+2)/(x^(2*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 20 2013
G.f.: conjecture: 1/(Q(0) - 1), where Q(k) = 1 + (-x)^k - (-x)^(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 25 2013
a(n) ~ exp(sqrt(n/2)*Pi)/(8*sqrt(2)*n). - Vaclav Kotesovec, Sep 26 2016
G.f.: Sum_{k>=0} x^(2*k) * Product_{j=1..k} (1+x^(2*j-1))/(1-x^(2*j)). - Seiichi Manyama, Jul 11 2018

A316718 Expansion of Product_{k=1..6} (1+x^(2*k-1))/(1-x^(2*k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 34, 41, 52, 65, 78, 93, 113, 137, 162, 189, 224, 266, 308, 355, 414, 480, 549, 626, 717, 820, 928, 1045, 1183, 1337, 1496, 1670, 1871, 2091, 2321, 2571, 2853, 3161, 3484, 3830, 4218, 4640, 5078, 5549, 6072, 6633, 7219
Offset: 0

Views

Author

Seiichi Manyama, Jul 11 2018

Keywords

Links

Crossrefs

Product_{k=1..b} (1+x^(2*k-1))/(1-x^(2*k)): A000012 (b=1), A004525(n+1) (b=2), A000933(n+5) (b=3), A089597 (b=4), A014670 (b=5), this sequence (b=6), A316719 (b=7), A316720 (b=8), A316721 (b=9), A316722 (b=10).
Cf. A316675.

Programs

  • PARI
    N=99; x='x+O('x^N); Vec(prod(k=1, 6, (1+x^(2*k-1))/(1-x^(2*k))))

A316719 Expansion of Product_{k=1..7} (1+x^(2*k-1))/(1-x^(2*k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 54, 68, 83, 100, 122, 149, 179, 212, 253, 303, 357, 417, 490, 575, 668, 772, 893, 1033, 1187, 1356, 1551, 1773, 2015, 2281, 2583, 2922, 3291, 3695, 4147, 4650, 5197, 5791, 6450, 7179, 7966, 8818, 9757, 10785, 11893
Offset: 0

Views

Author

Seiichi Manyama, Jul 11 2018

Keywords

Links

Crossrefs

Product_{k=1..b} (1+x^(2*k-1))/(1-x^(2*k)): A000012 (b=1), A004525(n+1) (b=2), A000933(n+5) (b=3), A089597 (b=4), A014670 (b=5), A316718 (b=6), this sequence (b=7), A316720 (b=8), A316721 (b=9), A316722 (b=10).
Cf. A316675.

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[(1 + x^(2 k - 1)) / (1 - x^(2 k)), {k, 1, 7}], {x, 0, nmax}], x] (* Vincenzo Librandi, Jul 12 2018 *)
  • PARI
    N=99; x='x+O('x^N); Vec(prod(k=1, 7, (1+x^(2*k-1))/(1-x^(2*k))))

A316720 Expansion of Product_{k=1..8} (1+x^(2*k-1))/(1-x^(2*k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 55, 70, 85, 103, 127, 156, 188, 224, 270, 326, 386, 454, 539, 638, 746, 869, 1016, 1186, 1372, 1581, 1827, 2108, 2415, 2758, 3156, 3605, 4094, 4639, 5261, 5956, 6715, 7553, 8499, 9552, 10694, 11950, 13357, 14908, 16589
Offset: 0

Views

Author

Seiichi Manyama, Jul 11 2018

Keywords

Links

Crossrefs

Product_{k=1..b} (1+x^(2*k-1))/(1-x^(2*k)): A000012 (b=1), A004525(n+1) (b=2), A000933(n+5) (b=3), A089597 (b=4), A014670 (b=5), A316718 (b=6), A316719 (b=7), this sequence (b=8), A316721 (b=9), A316722 (b=10).
Cf. A316675.

Programs

  • PARI
    N=99; x='x+O('x^N); Vec(prod(k=1, 8, (1+x^(2*k-1))/(1-x^(2*k))))

A316721 Expansion of Product_{k=1..9} (1+x^(2*k-1))/(1-x^(2*k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 55, 70, 86, 105, 129, 159, 193, 231, 279, 338, 403, 477, 568, 675, 795, 932, 1094, 1284, 1497, 1736, 2016, 2340, 2700, 3105, 3573, 4106, 4699, 5363, 6118, 6972, 7921, 8974, 10163, 11500, 12974, 14606, 16435, 18471
Offset: 0

Views

Author

Seiichi Manyama, Jul 11 2018

Keywords

Links

Crossrefs

Product_{k=1..b} (1+x^(2*k-1))/(1-x^(2*k)): A000012 (b=1), A004525(n+1) (b=2), A000933(n+5) (b=3), A089597 (b=4), A014670 (b=5), A316718 (b=6), A316719 (b=7), A316720 (b=8), this sequence (b=9), A316722 (b=10).
Cf. A316675.

Programs

  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1 + x^(2 k - 1)) / (1 - x^(2 k)), {k, 1, 9}], {x, 0, nmax}], x] (* Vincenzo Librandi, Jul 12 2018 *)
  • PARI
    N=99; x='x+O('x^N); Vec(prod(k=1, 9, (1+x^(2*k-1))/(1-x^(2*k))))

A316722 Expansion of Product_{k=1..10} (1+x^(2*k-1))/(1-x^(2*k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 55, 70, 86, 105, 130, 161, 195, 234, 284, 345, 412, 489, 585, 698, 824, 969, 1143, 1347, 1575, 1834, 2141, 2496, 2891, 3339, 3862, 4460, 5125, 5876, 6740, 7720, 8810, 10031, 11423, 12993, 14730, 16669, 18862, 21315
Offset: 0

Views

Author

Seiichi Manyama, Jul 11 2018

Keywords

Links

Crossrefs

Product_{k=1..b} (1+x^(2*k-1))/(1-x^(2*k)): A000012 (b=1), A004525(n+1) (b=2), A000933(n+5) (b=3), A089597 (b=4), A014670 (b=5), A316718 (b=6), A316719 (b=7), A316720 (b=8), A316721 (b=9), this sequence (b=10).
Cf. A316675.

Programs

  • PARI
    N=99; x='x+O('x^N); Vec(prod(k=1, 10, (1+x^(2*k-1))/(1-x^(2*k))))
Showing 1-6 of 6 results.

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

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