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

A369807 Expansion of 1/(1 - x^4/(1-x)^7).

Original entry on oeis.org

1, 0, 0, 0, 1, 7, 28, 84, 211, 476, 1029, 2276, 5384, 13594, 35371, 91667, 232681, 577710, 1413462, 3442498, 8414484, 20717963, 51346109, 127678961, 317496621, 787941379, 1950774874, 4821609252, 11910608942, 29432604429, 72787392898, 180131835001
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2024

Keywords

Comments

Number of compositions of 7*n-4 into parts 4 and 7.

Links

Crossrefs

Cf. A099253, A369805, A369806, A369808, A369809.
Cf. A369815.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/(1-x^4/(1-x)^7))
  • PARI
    a(n) = sum(k=0, n\4, binomial(n-1+3*k, n-4*k));

Formula

a(n) = A369815(7*n-4) for n > 0.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 34*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 7.
a(n) = Sum_{k=0..floor(n/4)} binomial(n-1+3*k,n-4*k).

A369805 Expansion of 1/(1 - x^2/(1-x)^7).

Original entry on oeis.org

1, 0, 1, 7, 29, 98, 316, 1043, 3536, 12083, 41168, 139750, 473824, 1607014, 5453022, 18506947, 62808496, 213144034, 723295969, 2454483506, 8329290739, 28265565587, 95919580313, 325504019213, 1104600373788, 3748469764612, 12720462563684, 43166996581876
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2024

Keywords

Comments

Number of compositions of 7*n-2 into parts 2 and 7.

Links

Crossrefs

Cf. A099253, A369806, A369807, A369808, A369809.
Cf. A369813.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-x^2/(1-x)^7))
  • PARI
    a(n) = sum(k=0, n\2, binomial(n-1+5*k, n-2*k));

Formula

a(n) = A369813(7*n-2) for n > 0.
a(n) = 7*a(n-1) - 20*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 7.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-1+5*k,n-2*k).

A369806 Expansion of 1/(1 - x^3/(1-x)^7).

Original entry on oeis.org

1, 0, 0, 1, 7, 28, 85, 224, 567, 1485, 4117, 11802, 33909, 96182, 269402, 750275, 2090728, 5845015, 16384908, 45973701, 128944042, 361364501, 1012168575, 2834690172, 7939970075, 22244001961, 62323608147, 174620915138, 489240430938, 1370662332271, 3839992876850
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2024

Keywords

Comments

Number of compositions of 7*n-3 into parts 3 and 7.

Links

Crossrefs

Cf. A099253, A369805, A369807, A369808, A369809.
Cf. A369814.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/(1-x^3/(1-x)^7))
  • PARI
    a(n) = sum(k=0, n\3, binomial(n-1+4*k, n-3*k));

Formula

a(n) = A369814(7*n-3) for n > 0.
a(n) = 7*a(n-1) - 21*a(n-2) + 36*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 7.
a(n) = Sum_{k=0..floor(n/3)} binomial(n-1+4*k,n-3*k).

A373931 Number of compositions of 7*n-5 into parts 1 and 7.

Original entry on oeis.org

1, 4, 17, 83, 413, 2037, 10010, 49183, 241722, 1188097, 5839638, 28702296, 141073905, 693388850, 3408058991, 16750869834, 82331801783, 404667078256, 1988969518921, 9775936716973, 48049473757425, 236166824233838, 1160777933797328, 5705311980035178
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2024

Keywords

Links

Crossrefs

Cf. A099253, A373907, A373928, A373929, A373930, A373932.
Cf. A005709, A369808.

Programs

  • Mathematica
    a[n_]:=n*(1 + n)*HypergeometricPFQ[{1-n,(2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6}, {3/7, 4/7, 5/7, 6/7, 8/7, 9/7}, -6^6/7^7]/2; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+1+6*k, n-1-k));

Formula

a(n) = A005709(7*n-5).
a(n) = Sum_{k=0..n} binomial(n+1+6*k,n-1-k).
a(n) = 8*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: x*(1-x)^4/((1-x)^7 - x).
a(n) = n*(1 + n)*hypergeom([1-n,(2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6], [3/7, 4/7, 5/7, 6/7, 8/7, 9/7], -6^6/7^7)/2. - Stefano Spezia, Jun 23 2024

A373937 Number of compositions of 7*n-5 into parts 6 and 7.

Original entry on oeis.org

0, 0, 0, 0, 1, 6, 21, 56, 126, 252, 463, 805, 1378, 2457, 4823, 10556, 24753, 58976, 137808, 310974, 675850, 1420916, 2914906, 5900631, 11931283, 24360194, 50559900, 106791426, 228638698, 492908713, 1062928750, 2281600816, 4862773227, 10287720750
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2024

Keywords

Links

Crossrefs

Cf. A369809, A373912, A373933, A373934, A373935, A373936.
Cf. A017847, A369808.

Programs

  • PARI
    a(n) = sum(k=0, n\6, binomial(n+k, n-5-6*k));

Formula

a(n) = A017847(7*n-5).
a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-5-6*k).
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).
G.f.: x^5*(1-x)/((1-x)^7 - x^6).
a(n) = A369809(n+1)-A369809(n). - R. J. Mathar, Jun 24 2024
Showing 1-6 of 6 results.

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