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.

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

A368475 Expansion of o.g.f. (1-x)^5/((1-x)^5 - x^4).

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 15, 35, 71, 136, 265, 550, 1211, 2732, 6126, 13485, 29191, 62648, 134408, 289656, 627401, 1363124, 2963186, 6434484, 13951852, 30221185, 65442625, 141745045, 307137901, 665732417, 1443184210, 3128438335, 6780867186, 14696002913, 31848721632
Offset: 0

Views

Author

Enrique Navarrete, Dec 26 2023

Keywords

Comments

For n > 0, a(n) is the number of ways to split [n] into an unspecified number of intervals and then choose 4 blocks (i.e., subintervals) from each interval. For example, for n=12, a(12)=1211 since the number of ways to split [12] into intervals and then select 4 blocks from each interval is C(12,4) + C(8,4)*C(4,4) + C(7,4)*C(5,4) + C(6,4)*C(6,4) + C(5,4)*C(7,4) + C(4,4)*C(8,4) + C(4,4)*C(4,4)*C(4,4) for a total of 1211 ways.
For n > 0, a(n) is also the number of compositions of n using parts of size at least 4 where there are binomial(i,4) types of i, i >= 4 (see example).
Number of compositions of 5*n-4 into parts 4 and 5. - Seiichi Manyama, Feb 01 2024

Examples

			Since there are C(4,4) = 1 type of 4, C(5,4) = 5 types of 5, C(6,4) = 15 types of 6, C(7,4) = 35 types of 7, C(8,4) = 70 types of 8, and (12,4) = 495 types of 12, we can write 12 in the following ways:
  12: 495 ways;
  8+4: 70 ways;
  7+5: 175 ways;
  6+6: 225 ways;
  5+7: 175 ways;
  4+8: 70 ways;
  4+4+4: 1 way, for a total of 1211 ways.

Links

Crossrefs

Cf. A000332, A017827, A099131, A290998.
Cf. A079675, A369803, A369804.
Cf. A088305, A095263, A290998, A369794, A369809.

Programs

  • Mathematica
    CoefficientList[Series[(1 - x)^5/((1 - x)^5 - x^4), {x, 0, 50}], x] (* Wesley Ivan Hurt, Dec 26 2023 *)
  • PARI
    Vec((1-x)^5/((1-x)^5 - x^4) + O(x^40)) \\ Michel Marcus, Dec 27 2023

Formula

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 4*a(n-4) + a(n-5), n>=6; a(0)=1, a(1)=a(2)=a(3)=0, a(4)=1, a(5)=5.
G.f.: 1/(1-Sum_{k>=4} binomial(k,4)*x^k).
G.f.: 1/p(S), where p(S) = 1 - S^4 - S^5 and S = x/(1-x).
First differences of A099131. - R. J. Mathar, Jan 29 2024
a(n) = A017827(5*n-4) = Sum_{k=0..floor((5*n-4)/4)} binomial(k,5*n-4-4*k) for n > 0. - Seiichi Manyama, Feb 01 2024
a(n) = Sum_{k=0..floor(n/4)} binomial(n-1+k,n-4*k). - Seiichi Manyama, Feb 02 2024

A373961 Number of compositions of 6*n-1 into parts 5 and 6.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 15, 44, 129, 340, 804, 1742, 3550, 7009, 13835, 28033, 58993, 128136, 282569, 622575, 1357136, 2918449, 6204578, 13104675, 27646776, 58502733, 124411595, 265807567, 569552644, 1221316021, 2616456236, 5595314908, 11944318042, 25466629978
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2024

Keywords

Links

Crossrefs

Cf. A107025, A369794, A373962, A373963, A373964.
Cf. A017837.

Programs

  • PARI
    a(n) = sum(k=0, n\5, binomial(n+k, n-1-5*k));

Formula

a(n) = A017837(6*n-1).
a(n) = Sum_{k=0..floor(n/5)} binomial(n+k,n-1-5*k).
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).
G.f.: x*(1-x)^4/((1-x)^6 - x^5).
a(n) = A373962(n+1) - A373962(n).

A373962 Number of compositions of 6*n-2 into parts 5 and 6.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 22, 37, 81, 210, 550, 1354, 3096, 6646, 13655, 27490, 55523, 114516, 242652, 525221, 1147796, 2504932, 5423381, 11627959, 24732634, 52379410, 110882143, 235293738, 501101305, 1070653949, 2291969970, 4908426206, 10503741114, 22448059156
Offset: 1

Views

Author

Seiichi Manyama, Jun 24 2024

Keywords

Links

Crossrefs

Cf. A107025, A369794, A373961, A373963, A373964.
Cf. A017837.

Programs

  • Mathematica
    LinearRecurrence[{6,-15,20,-15,7,-1},{0,1,3,6,10,15},40] (* Harvey P. Dale, Nov 16 2024 *)
  • PARI
    a(n) = sum(k=0, n\5, binomial(n+k, n-2-5*k));

Formula

a(n) = A017837(6*n-2).
a(n) = Sum_{k=0..floor(n/5)} binomial(n+k,n-2-5*k).
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).
G.f.: x^2*(1-x)^3/((1-x)^6 - x^5).
a(n) = A373963(n+1) - A373963(n).

A373963 Number of compositions of 6*n-3 into parts 5 and 6.

Original entry on oeis.org

0, 0, 1, 4, 10, 20, 35, 57, 94, 175, 385, 935, 2289, 5385, 12031, 25686, 53176, 108699, 223215, 465867, 991088, 2138884, 4643816, 10067197, 21695156, 46427790, 98807200, 209689343, 444983081, 946084386, 2016738335, 4308708305, 9217134511, 19720875625
Offset: 1

Views

Author

Seiichi Manyama, Jun 24 2024

Keywords

Links

Crossrefs

Cf. A107025, A369794, A373961, A373962, A373964.
Cf. A017837.

Programs

  • PARI
    a(n) = sum(k=0, n\5, binomial(n+k, n-3-5*k));

Formula

a(n) = A017837(6*n-3).
a(n) = Sum_{k=0..floor(n/5)} binomial(n+k,n-3-5*k).
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).
G.f.: x^3*(1-x)^2/((1-x)^6 - x^5).
a(n) = A373964(n+1) - A373964(n).

A373964 Number of compositions of 6*n-4 into parts 5 and 6.

Original entry on oeis.org

0, 0, 0, 1, 5, 15, 35, 70, 127, 221, 396, 781, 1716, 4005, 9390, 21421, 47107, 100283, 208982, 432197, 898064, 1889152, 4028036, 8671852, 18739049, 40434205, 86861995, 185669195, 395358538, 840341619, 1786426005, 3803164340, 8111872645, 17329007156
Offset: 1

Views

Author

Seiichi Manyama, Jun 24 2024

Keywords

Links

Crossrefs

Cf. A107025, A369794, A373961, A373962, A373963.
Cf. A017837.

Programs

  • PARI
    a(n) = sum(k=0, n\5, binomial(n+k, n-4-5*k));

Formula

a(n) = A017837(6*n-4).
a(n) = Sum_{k=0..floor(n/5)} binomial(n+k,n-4-5*k).
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).
G.f.: x^4*(1-x)/((1-x)^6 - x^5).
a(n) = A369794(n+1) - A369794(n).
Showing 1-6 of 6 results.

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