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 1-9 of 9 results.

A128425 Primes in A000933.

Original entry on oeis.org

2, 3, 5, 11, 13, 23, 29, 43, 59, 73, 83, 137, 151, 173, 181, 239, 257, 347, 463, 653, 683, 809, 859, 1093, 1151, 1171, 1231, 1553, 1669, 1693, 1789, 1913, 2311, 2423, 2837, 3121, 3251, 3623, 3943, 4583, 4861, 5023, 5483, 5569, 5743, 6053
Offset: 1

Views

Author

Jonathan Vos Post, May 04 2007

Keywords

Crossrefs

Cf. A000933.

A014670 G.f.: (1+x)*(1+x^3)*(1+x^5)*(1+x^7)*(1+x^9)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)*(1-x^10)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 20, 26, 32, 38, 47, 58, 69, 81, 96, 114, 133, 153, 177, 206, 236, 267, 304, 346, 390, 437, 490, 550, 613, 679, 753, 835, 921, 1011, 1111, 1221, 1335, 1455, 1586, 1728, 1877, 2032, 2200, 2382, 2571, 2768, 2980, 3207, 3443, 3689, 3952
Offset: 0

Views

Author

N. J. A. Sloane, Dec 31 2003

Keywords

Comments

Poincaré series [or Poincare series] (or Molien series) for symmetric invariants in F_2(b_1, b_2, ... b_n) ⊗ E(e_1, e_2, ... e_n) with b_i 2-dimensional, e_i one-dimensional and the permutation action of S_n, in the case n=5.

References

  • A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 108.

Links

Crossrefs

Cf. A006950, A000933, A089597.

Programs

  • Mathematica
    CoefficientList[Series[(1+x)*(1+x^3)*(1+x^5)*(1+x^7)*(1+x^9)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)*(1-x^10)), {x, 0, 50}], x] (* Jinyuan Wang, Mar 10 2020 *)
LinearRecurrence[{3,-5,8,-11,14,-18,21,-23,24,-24,23,-21,18,-14,11,-8,5,-3,1},{1,1,1,2,3,4,5,7,10,13,16,20,26,32,38,47,58,69,81},60] (* Harvey P. Dale, Mar 28 2023 *)
  • PARI
    Vec((1+x)*(1+x^3)*(1+x^5)*(1+x^7)*(1+x^9)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)*(1-x^10))+ O(x^100)) \\ Michel Marcus, Mar 18 2014

Formula

G.f.: -(x^2-x+1) *(x^6-x^5+x^4-x^3+x^2-x+1) *(x^6-x^3+1) / ( (x^4+x^3+x^2+x+1) *(1+x+x^2) *(x^4+1) *(x^2+1)^2 *(x-1)^5 ). - R. J. Mathar, Dec 18 2014

A316675 Triangle read by rows: T(n,k) gives the number of ways to stack n triangles in a valley so that the right wall has k triangles for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Jul 10 2018

Keywords

Examples

			T(8,4) = 3.
    *                             *
   / \                           / \
  *---*   *     *---*---*       *---*
   \ / \ / \     \ / \ / \     / \ / \
    *---*---*     *---*---*   *---*---*
     \ / \ /       \ / \ /     \ / \ /
      *---*         *---*       *---*
       \ /           \ /         \ /
        *             *           *
Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 1;
  0, 0, 1, 1, 1;
  0, 0, 1, 1, 1, 1;
  0, 0, 1, 1, 1, 1,  1;
  0, 0, 1, 1, 2, 1,  1,  1;
  0, 0, 1, 1, 3, 2,  1,  1,  1;
  0, 0, 1, 1, 3, 3,  2,  1,  1,  1;
  0, 0, 1, 1, 3, 3,  3,  2,  1,  1,  1;
  0, 0, 1, 1, 4, 3,  4,  3,  2,  1,  1, 1;
  0, 0, 1, 1, 5, 4,  5,  4,  3,  2,  1, 1, 1;
  0, 0, 1, 1, 5, 5,  6,  5,  4,  3,  2, 1, 1, 1;
  0, 0, 1, 1, 5, 5,  8,  6,  5,  4,  3, 2, 1, 1, 1;
  0, 0, 1, 1, 6, 5, 10,  8,  7,  5,  4, 3, 2, 1, 1, 1;
  0, 0, 1, 1, 7, 6, 11, 10, 10,  7,  5, 4, 3, 2, 1, 1, 1;
  0, 0, 1, 1, 7, 7, 13, 11, 12, 10,  7, 5, 4, 3, 2, 1, 1, 1;
  0, 0, 1, 1, 7, 7, 16, 13, 14, 12, 10, 7, 5, 4, 3, 2, 1, 1, 1;
  ...

Links

Crossrefs

Row sums give A006950.
Sums of even columns give A059777.
Cf. A004525, A000933, A089597, A014670, A316718, A316719, A316720, A316721, A316722.
Cf. A072233.

Formula

For m >= 0,
Sum_{n>=2m} T(n,2m) *x^n = x^(2m) * Product_{j=1..m} (1+x^(2j-1))/(1-x^(2j)).
Sum_{n>=2m+1} T(n,2m+1)*x^n = x^(2m+1) * Product_{j=1..m} (1+x^(2j-1))/(1-x^(2j)).

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))))

A253012 a(n) = ceiling( (n+1) * (n+2) / 12).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 6, 8, 10, 11, 13, 16, 18, 20, 23, 26, 29, 32, 35, 39, 43, 46, 50, 55, 59, 63, 68, 73, 78, 83, 88, 94, 100, 105, 111, 118, 124, 130, 137, 144, 151, 158, 165, 173, 181, 188, 196, 205, 213, 221, 230, 239, 248, 257, 266, 276, 286, 295, 305
Offset: 0

Views

Author

Michael Somos, Dec 25 2014

Keywords

Examples

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 8*x^8 + 10*x^9 + ...

Links

Crossrefs

Cf. A000933, A091972.

Programs

  • Magma
    [Ceiling((n+1)*(n+2)/12): n in [0..60]]; // G. C. Greubel, Aug 04 2018
  • Mathematica
    a[ n_] := Ceiling[ (n+1) (n+2) / 12];
  • PARI
    {a(n) = ceil( (n+1) * (n+2) / 12)};

Formula

Euler transform of length 10 sequence [ 1, 0, 1, 1, 1, 0, 0, 0, 0, -1].
G.f.: (1 + x^5) / ((1 - x) * (1 - x^3) * (1 - x^4)) = (1-x+x^2-x^3+x^4) /( (1+x+x^2) *(1+x^2) *(1-x)^3 ).
a(n) = a(-3-n) for all n in Z.
a(n) = A000933(n+5) for all n>=-2.
a(n) - a(n-1) = A091972(n) for all n in Z.
a(n) - 2*a(n+1) + a(n+2) is a period 12 integer sequence.
a(n) - 2*a(n+1) + 2*a(n+2) - 2*a(n+3) + a(n+4) = 1 if n == 1 (mod 3) else 0.
Showing 1-9 of 9 results.

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

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