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.

A371777 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n+2,n-3*k).

Original entry on oeis.org

1, 4, 15, 57, 220, 858, 3368, 13276, 52479, 207861, 824527, 3274395, 13015081, 51769813, 206045841, 820475513, 3268499356, 13025237058, 51922543076, 207034128448, 825713206746, 3293865399518, 13142007903586, 52443095356218, 209304385553096, 835459642193284
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2024

Keywords

Crossrefs

Cf. A371778, A371779, A371780.
Cf. A144904, A371758, A371773.
Cf. A032443.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n+2, n-3*k));

Formula

a(n) = [x^n] 1/(((1-x)^3-x^3) * (1-x)^n).
a(n) = binomial(2*(n+1), n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [1+n/3, (4+n)/3, (5+n)/3], -1). - Stefano Spezia, Apr 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: n*a(n) = 3*(3*n-2)*a(n-1) - 6*(4*n-5)*a(n-2) + 8*(2*n-3)*a(n-3).
G.f.: (1 + sqrt(1-4*x))/(2*(1-x)*(1-4*x)).
a(n) ~ 2^(2*n+1)/3. (End)

A371758 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-3*k-1,n-3*k).

Original entry on oeis.org

1, 1, 3, 11, 39, 141, 519, 1933, 7263, 27479, 104543, 399543, 1532779, 5899167, 22766607, 88073091, 341425551, 1326019653, 5158412943, 20096457549, 78396460299, 306190920837, 1197181197567, 4685523856881, 18354865147011, 71962695111841, 282357198103815
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2024

Keywords

Crossrefs

Cf. A371770, A371771, A371772.
Cf. A144904, A371773, A371777.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n-3*k-1, n-3*k));

Formula

a(n) = [x^n] 1/((1-x^3) * (1-x)^n).
a(n) = binomial(2*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [(1-2*n)/3, 2*(1-n)/3, 1-2*n/3], 1). - Stefano Spezia, Apr 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: 3*n*(7*n-11)*a(n) = 6*(2*n-3)*(7*n-4)*a(n-1) - n*(7*n-11)*a(n-2) + 2*(2*n-3)*(7*n-4)*a(n-3).
a(n) ~ 2^(2*n+2) / (7*sqrt(Pi*n)). (End)

A371774 a(n) = Sum_{k=0..floor(n/3)} binomial(3*n-k+1,n-3*k).

Original entry on oeis.org

1, 4, 21, 121, 727, 4473, 27949, 176549, 1124332, 7205511, 46411744, 300183757, 1948255421, 12681654613, 82755728730, 541213820732, 3546268982757, 23276100962571, 153004515241866, 1007131032951572, 6637396253259291, 43791520333601111
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2024

Keywords

Crossrefs

Cf. A371773, A371775, A371776.
Cf. A371754, A371770.
Cf. A045721.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(3*n-k+1, n-3*k));

Formula

a(n) = [x^n] 1/(((1-x)^2-x^3) * (1-x)^(2*n)).
a(n) = binomial(1+3*n, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [-1-3*n, 1+n, 3/2+n], 27/4). - Stefano Spezia, Apr 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: 2*n*(2*n - 1)*(671*n^4 - 4757*n^3 + 11743*n^2 - 11533*n + 3516)*a(n) = (44957*n^6 - 350256*n^5 + 997889*n^4 - 1236792*n^3 + 563834*n^2 + 39768*n - 60480)*a(n-1) - 10*(19459*n^6 - 156741*n^5 + 461272*n^4 - 575421*n^3 + 211099*n^2 + 106572*n - 60480)*a(n-2) + (93269*n^6 - 753150*n^5 + 2221631*n^4 - 2772678*n^3 + 999800*n^2 + 543408*n - 302400)*a(n-3) - 3*(3*n - 8)*(3*n - 7)*(671*n^4 - 2073*n^3 + 1498*n^2 + 366*n - 360)*a(n-4).
a(n) ~ 3^(3*n + 5/2) / (11 * sqrt(Pi*n) * 2^(2*n)). (End)

A371775 a(n) = Sum_{k=0..floor(n/3)} binomial(4*n-k+1,n-3*k).

Original entry on oeis.org

1, 5, 36, 287, 2396, 20539, 179125, 1581282, 14085997, 126357958, 1139825257, 10328791996, 93951594230, 857328996139, 7844767641718, 71951952863375, 661311093597592, 6089245462608316, 56160004711457917, 518707264791838694, 4797177987838607105
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2024

Keywords

Crossrefs

Cf. A371773, A371774, A371776.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(4*n-k+1, n-3*k));

Formula

a(n) = [x^n] 1/(((1-x)^2-x^3) * (1-x)^(3*n)).

A371776 a(n) = Sum_{k=0..floor(n/3)} binomial(5*n-k+1,n-3*k).

Original entry on oeis.org

1, 6, 55, 561, 6005, 66080, 740342, 8400074, 96206994, 1109874635, 12877808194, 150122945518, 1756887201266, 20628519611407, 242891806678851, 2866906127955287, 33910670558191711, 401857349039547372, 4770115555036932777, 56706219260783415643
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2024

Keywords

Crossrefs

Cf. A371773, A371774, A371775.

Programs

  • Mathematica
    Table[Sum[Binomial[5n-k+1,n-3k],{k,0,Floor[n/3]}],{n,0,20}] (* Harvey P. Dale, May 13 2025 *)
  • PARI
    a(n) = sum(k=0, n\3, binomial(5*n-k+1, n-3*k));

Formula

a(n) = [x^n] 1/(((1-x)^2-x^3) * (1-x)^(4*n)).

A371819 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-k+1,n-3*k).

Original entry on oeis.org

1, 3, 10, 34, 118, 417, 1497, 5447, 20047, 74493, 279054, 1052467, 3992204, 15216662, 58239175, 223688159, 861769598, 3328779906, 12887832493, 49998248601, 194315972151, 756406944446, 2948649839743, 11509316352548, 44976030493706, 175942932935325
Offset: 0

Views

Author

Seiichi Manyama, Apr 06 2024

Keywords

Crossrefs

Cf. A001700, A120305, A371818, A371820.
Cf. A371773.

Programs

  • PARI
    a(n) = sum(k=0, n\3, (-1)^k*binomial(2*n-k+1, n-3*k));

Formula

a(n) = [x^n] 1/(((1-x)^2+x^3) * (1-x)^n).
a(n) = binomial(1+2*n, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [-1-2*n, 1+n/2, (3+n)/2], -27/4). - Stefano Spezia, Apr 07 2024

A371854 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-k+2,n-3*k).

Original entry on oeis.org

1, 4, 15, 57, 219, 847, 3290, 12819, 50066, 195909, 767790, 3013002, 11837043, 46548919, 183209125, 721628692, 2844297119, 11217639757, 44265835891, 174765349896, 690308413773, 2727823240762, 10783518961394, 42644560775835, 168699835910561, 667580653569309
Offset: 0

Views

Author

Seiichi Manyama, Apr 09 2024

Keywords

Crossrefs

Cf. A002450, A105872, A371842.
Cf. A360150, A371773.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n-k+2, n-3*k));

Formula

a(n) = [x^n] 1/(((1-x)^2-x^3) * (1-x)^(n+1)).
Showing 1-7 of 7 results.

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