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.

A369809 Expansion of 1/(1 - x^6/(1-x)^7).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 7, 28, 84, 210, 462, 925, 1730, 3108, 5565, 10388, 20944, 45697, 104673, 242481, 553455, 1229305, 2650221, 5565127, 11465758, 23397041, 47757235, 98317135, 205108561, 433747259, 926655972, 1989584722, 4271185538, 9133958765, 19421679515
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2024

Keywords

Comments

Number of compositions of 7*n-6 into parts 6 and 7.

Links

Crossrefs

Cf. A099253, A369805, A369806, A369807, A369808.
Cf. A088305, A095263, A290998, A368475, A369794.
Cf. A000579, A017847.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/(1-x^6/(1-x)^7))
  • PARI
    a(n) = sum(k=0, n\6, binomial(n-1+k, n-6*k));

Formula

G.f. (1-x)^7/((1-x)^7-x^6).
a(n) = A017847(7*n-6) = Sum_{k=0..floor((7*n-6)/6)} binomial(k,7*n-6-6*k) for n > 0.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 6*a(n-6) + a(n-7) for n > 7.
a(n) = Sum_{k=0..floor(n/6)} binomial(n-1+k,n-6*k).
a(n) = A373912(n)-A373912(n-1). - R. J. Mathar, Jun 24 2024

A369804 Expansion of 1/(1 - x^3/(1-x)^5).

Original entry on oeis.org

1, 0, 0, 1, 5, 15, 36, 80, 181, 431, 1060, 2617, 6401, 15521, 37513, 90741, 219918, 533619, 1295022, 3141826, 7619870, 18478155, 44810670, 108676262, 263576791, 639267800, 1550434777, 3760269946, 9119740067, 22118021213, 53642768716, 130099857234, 315531401964
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2024

Keywords

Comments

Number of compositions of 5*n-3 into parts 3 and 5.

Links

Crossrefs

Cf. A079675, A368475, A369803.
Cf. A369845, A369846, A369847, A369848.
Cf. A052920.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/(1-x^3/(1-x)^5))
  • PARI
    a(n) = sum(k=0, n\3, binomial(n-1+2*k, n-3*k));

Formula

a(n) = A052920(5*n-3) for n > 0.
a(n) = 5*a(n-1) - 10*a(n-2) + 11*a(n-3) - 5*a(n-4) + a(n-5) for n > 5.
a(n) = Sum_{k=0..floor(n/3)} binomial(n-1+2*k,n-3*k).
a(n) = A369845(n) - A369845(n-1). - R. J. Mathar, Feb 14 2024

A369803 Expansion of 1/(1 - x^2/(1-x)^5).

Original entry on oeis.org

1, 0, 1, 5, 16, 45, 126, 361, 1046, 3032, 8771, 25348, 73252, 211724, 612009, 1769080, 5113647, 14781237, 42725841, 123501151, 356986401, 1031887518, 2982723523, 8621714049, 24921502864, 72036871920, 208226244217, 601888555723, 1739789499591, 5028950081882
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2024

Keywords

Comments

Number of compositions of 5*n-2 into parts 2 and 5.

Links

Crossrefs

Cf. A079675, A368475, A369804.
Cf. A369840, A369842, A369843, A369844.
Cf. A001687.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-x^2/(1-x)^5))
  • PARI
    a(n) = sum(k=0, n\2, binomial(n-1+3*k, n-2*k));

Formula

a(n) = A001687(5*n-1) for n > 0.
a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-1+3*k,n-2*k).
a(n) = A369840(n)-A369840(n-1). - R. J. Mathar, Feb 14 2024

A369794 Expansion of 1/(1 - x^5/(1-x)^6).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 6, 21, 56, 126, 253, 474, 870, 1651, 3367, 7372, 16762, 38183, 85290, 185573, 394555, 826752, 1724816, 3613968, 7642004, 16313856, 35052905, 75487110, 162349105, 348018300, 743376838, 1583718457, 3370144462, 7173308802, 15285181447
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2024

Keywords

Comments

Number of compositions of 6*n-5 into parts 5 and 6.

Links

Crossrefs

Cf. A095263, A290998, A368475.
Cf. A000389, A099242, A017837, A107025.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/(1-x^5/(1-x)^6))

Formula

a(n) = A107025(n)-A107025(n-1). First differences of A107025.
a(n) = A017837(6*n-5) = Sum_{k=0..floor((6*n-5)/5)} binomial(k,6*n-5-5*k) for n > 0.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 7*a(n-5) - a(n-6) for n > 6.
a(n) = Sum_{k=0..floor(n/5)} binomial(n-1+k,n-5*k).

A369849 Number of compositions of 5*n-1 into parts 4 and 5.

Original entry on oeis.org

1, 2, 3, 4, 6, 13, 35, 92, 220, 484, 1013, 2092, 4382, 9404, 20552, 45185, 99009, 215481, 466361, 1006897, 2174834, 4705895, 10200142, 22128873, 48009456, 104111224, 225655617, 488945055, 1059372394, 2295532150, 4974876116, 10782658417, 23371307904, 50655960304
Offset: 1

Views

Author

Seiichi Manyama, Feb 03 2024

Keywords

Links

Crossrefs

Cf. A099131, A368475, A369850, A369851.
Cf. A017827.

Programs

  • Mathematica
    LinearRecurrence[{5, -10, 10, -4, 1}, {1, 2, 3, 4, 6}, 50] (* Paolo Xausa, Mar 15 2024 *)
  • PARI
    a(n) = sum(k=0, n\4, binomial(n+k, n-1-4*k));

Formula

a(n) = A017827(5*n-1).
a(n) = Sum_{k=0..floor(n/4)} binomial(n+k,n-1-4*k).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 4*a(n-4) + a(n-5).
G.f.: x*(1-x)^3/((1-x)^5 - x^4).

A369850 Number of compositions of 5*n-2 into parts 4 and 5.

Original entry on oeis.org

0, 1, 3, 6, 10, 16, 29, 64, 156, 376, 860, 1873, 3965, 8347, 17751, 38303, 83488, 182497, 397978, 864339, 1871236, 4046070, 8751965, 18952107, 41080980, 89090436, 193201660, 418857277, 907802332, 1967174726, 4262706876, 9237582992, 20020241409, 43391549313
Offset: 1

Views

Author

Seiichi Manyama, Feb 03 2024

Keywords

Links

Crossrefs

Cf. A099131, A368475, A369849, A369851.
Cf. A017827.

Programs

  • Mathematica
    LinearRecurrence[{5, -10, 10, -4, 1}, {0, 1, 3, 6, 10}, 50] (* Paolo Xausa, Mar 15 2024 *)
  • PARI
    a(n) = sum(k=0, n\4, binomial(n+k, n-2-4*k));

Formula

a(n) = A017827(5*n-2).
a(n) = Sum_{k=0..floor(n/4)} binomial(n+k,n-2-4*k).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 4*a(n-4) + a(n-5).
G.f.: x^2*(1-x)^2/((1-x)^5 - x^4).

A369851 Number of compositions of 5*n-3 into parts 4 and 5.

Original entry on oeis.org

0, 0, 1, 4, 10, 20, 36, 65, 129, 285, 661, 1521, 3394, 7359, 15706, 33457, 71760, 155248, 337745, 735723, 1600062, 3471298, 7517368, 16269333, 35221440, 76302420, 165392856, 358594516, 777451793, 1685254125, 3652428851, 7915135727, 17152718719, 37172960128
Offset: 1

Views

Author

Seiichi Manyama, Feb 03 2024

Keywords

Links

Crossrefs

Cf. A099131, A368475, A369849, A369850.
Cf. A017827.

Programs

  • Mathematica
    LinearRecurrence[{5, -10, 10, -4, 1}, {0, 0, 1, 4, 10}, 50] (* Paolo Xausa, Mar 15 2024 *)
  • PARI
    a(n) = sum(k=0, n\4, binomial(n+k, n-3-4*k));

Formula

a(n) = A017827(5*n-3).
a(n) = Sum_{k=0..floor(n/4)} binomial(n+k,n-3-4*k).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 4*a(n-4) + a(n-5).
G.f.: x^3*(1-x)/((1-x)^5 - x^4).
Showing 1-7 of 7 results.

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