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.

A361277 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(k*j,n-j)/j!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 7, 1, 1, 1, 7, 19, 25, 1, 1, 1, 9, 37, 97, 81, 1, 1, 1, 11, 61, 241, 581, 331, 1, 1, 1, 13, 91, 481, 1981, 3661, 1303, 1, 1, 1, 15, 127, 841, 4881, 17551, 26335, 5937, 1, 1, 1, 17, 169, 1345, 10001, 55321, 171697, 202049, 26785, 1
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2023

Keywords

Examples

			Square array begins:
  1,  1,   1,    1,    1,     1, ...
  1,  1,   1,    1,    1,     1, ...
  1,  3,   5,    7,    9,    11, ...
  1,  7,  19,   37,   61,    91, ...
  1, 25,  97,  241,  481,   841, ...
  1, 81, 581, 1981, 4881, 10001, ...

Crossrefs

Columns k=0..4 give A000012, A047974, A361278, A361279, A361280.
Main diagonal gives A361281.
Cf. A293012.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n, binomial(k*j, n-j)/j!);

Formula

E.g.f. of column k: exp(x * (1+x)^k).
T(0,k) = 1; T(n,k) = (n-1)! * Sum_{j=1..n} j * binomial(k,j-1) * T(n-j,k)/(n-j)!.

A361568 Expansion of e.g.f. exp(x^3/6 * (1+x)^3).

Original entry on oeis.org

1, 0, 0, 1, 12, 60, 130, 420, 8400, 101080, 781200, 4435200, 37714600, 607807200, 8660652000, 94007313400, 914497584000, 11566931376000, 198256136478400, 3275456501116800, 46558791351072000, 636647461257808000, 10238792220969312000, 194852563745775936000
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2023

Keywords

Crossrefs

Cf. A047974, A361567, A361569.
Cf. A115055, A361279.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/6*(1+x)^3)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/6*sum(j=3, i, j*binomial(3, j-3)*v[i-j+1]/(i-j)!)); v;

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(3*k,n-3*k)/(6^k * k!).
a(0) = 1; a(n) = ((n-1)!/6) * Sum_{k=3..n} k * binomial(3,k-3) * a(n-k)/(n-k)!.
a(n) = (n-1)*(n-2)/6 * (3*a(n-3) + 12*(n-3)*a(n-4) + 15*(n-3)*(n-4)*a(n-5) + 6*(n-3)*(n-4)*(n-5)*a(n-6)). -Seiichi Manyama, Jun 16 2024

A377964 Expansion of e.g.f. (1+x) * exp(x*(1+x)^3).

Original entry on oeis.org

1, 2, 9, 58, 389, 3186, 29437, 294554, 3233673, 38350594, 484794641, 6522118362, 92857444429, 1390937221298, 21858658599429, 359271578140666, 6156249977141777, 109722278546645634, 2029772196329985433, 38893956306343711994, 770622936760496106261, 15763542538016019828082
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2024

Keywords

Crossrefs

Cf. A018191, A377963.
Cf. A361279, A377966, A377967.

Programs

  • PARI
    a(n, s=1, t=3) = n!*sum(k=0, n, binomial(t*k+s, n-k)/k!);

Formula

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

A377966 Expansion of e.g.f. (1+x)^2 * exp(x*(1+x)^3).

Original entry on oeis.org

1, 3, 13, 85, 621, 5131, 48553, 500613, 5590105, 67453651, 868300581, 11854859413, 171122864773, 2598083998875, 41331779697601, 687151457132101, 11904595227392433, 214378528158055843, 4004773210169606845, 77459628036613435221, 1548502062887370346141
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2024

Keywords

Crossrefs

Cf. A377954, A377965.
Cf. A361279, A377964, A377967.

Programs

  • PARI
    a(n, s=2, t=3) = n!*sum(k=0, n, binomial(t*k+s, n-k)/k!);

Formula

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

A361571 Expansion of e.g.f. exp( (x * (1+x))^3 ).

Original entry on oeis.org

1, 0, 0, 6, 72, 360, 1080, 15120, 302400, 3689280, 32659200, 359251200, 6965481600, 133880947200, 2070484416000, 30305353478400, 559684629504000, 12582442768896000, 271843009108070400, 5401042458152140800, 111578968350001152000, 2657164887872022528000
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2023

Keywords

Links

Crossrefs

Cf. A047974, A361570.
Cf. A361279.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[(x(1+x))^3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 06 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x*(1+x))^3)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, j*binomial(3, j-3)*v[i-j+1]/(i-j)!)); v;

Formula

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

A377967 Expansion of e.g.f. (1+x)^3 * exp(x*(1+x)^3).

Original entry on oeis.org

1, 4, 19, 124, 961, 8236, 79339, 840484, 9595009, 117764596, 1542837091, 21406165804, 313381177729, 4822681240924, 77704955681851, 1307128152596116, 22899018541506049, 416756647023727204, 7863586717014612019, 153550319029835965276, 3097694623619639050561
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2024

Keywords

Crossrefs

Cf. A361279, A377964, A377966.
Cf. A351767.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[(1+x)^3 Exp[x*(1+x)^3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 28 2025 *)
  • PARI
    a(n, s=3, t=3) = n!*sum(k=0, n, binomial(t*k+s, n-k)/k!);

Formula

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

A373720 Expansion of e.g.f. exp(x * (1 + x^2)^3).

Original entry on oeis.org

1, 1, 1, 19, 73, 541, 5761, 35911, 515089, 5399353, 61253281, 991270171, 11862564121, 203249068309, 3295367161633, 52595413358671, 1060046073787681, 18422593177204081, 383150483373313729, 8042585703164409763, 165930214242407069161, 3968988522451484425741
Offset: 0

Views

Author

Seiichi Manyama, Jun 15 2024

Keywords

Crossrefs

Cf. A361279, A373718.

Programs

  • PARI
    a(n) = n!*sum(k=0, 3*n\7, binomial(3*n-6*k, k)/(n-2*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(3*n/7)} binomial(3*n-6*k,k)/(n-2*k)!.
a(n) == 1 (mod 18).
a(n) = a(n-1) + 9*(n-1)*(n-2)*a(n-3) + 15*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5) + 7*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*a(n-7).
Showing 1-7 of 7 results.

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