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 1-9 of 9 results.

A081126 Duplicate of A018191.

Original entry on oeis.org

1, 2, 5, 16, 53, 206, 817, 3620, 16361, 80218, 401501, 2139512, 11641885, 66599846, 388962953, 2367284236, 14700573137, 94523836850, 619674301621, 4186249123808, 28809504493061, 203556335785342, 1463877667140065
Offset: 0

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Author

Keywords

A377954 a(n) = n! * Sum_{k=0..n} binomial(k+2,n-k) / k!.

Original entry on oeis.org

1, 3, 9, 31, 117, 471, 2053, 9339, 45321, 227467, 1203681, 6556023, 37316029, 217944351, 1321360797, 8201728531, 52577120913, 344433580179, 2321103364921, 15960060854607, 112534486969221, 808555930139623, 5942117054417589, 44446333314841131
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2024

Keywords

Crossrefs

Cf. A018191, A047974, A377955, A377956.

Programs

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

Formula

E.g.f.: (1 + x)^2 * exp(x + x^2).
a(n) = -(n-4)*a(n-1) + 3*(n-1)*a(n-2) + 2*(n-1)*(n-2)*a(n-3) for n > 2.
a(n) = ((n^2-7*n+3)*a(n-1) + 2*(n-1)*(n^2-3*n-1)*a(n-2))/(n^2-5*n+3) for n > 1.
a(n) ~ n^(n/2 + 1) * 2^(n/2 - 3/2) / exp(1/8 - sqrt(n/2) + n/2) * (1 + 157/(48*sqrt(2*n))). - Vaclav Kotesovec, Nov 12 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!.

A377955 a(n) = n! * Sum_{k=0..n} binomial(k+3,n-k) / k!.

Original entry on oeis.org

1, 4, 15, 58, 241, 1056, 4879, 23710, 120033, 635356, 3478351, 19796514, 115988305, 703052728, 4372581711, 28022140486, 183804777409, 1238244635700, 8520907808143, 60061024788106, 431735704061361, 3171780156493264, 23730347517489295, 181115025566445678
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2024

Keywords

Crossrefs

Cf. A018191, A047974, A377954, A377956.

Programs

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

Formula

E.g.f.: (1 + x)^3 * exp(x + x^2).
a(n) = -(n-5)*a(n-1) + 3*(n-1)*a(n-2) + 2*(n-1)*(n-2)*a(n-3) for n > 2.

A377963 Expansion of e.g.f. (1+x) * exp(x*(1+x)^2).

Original entry on oeis.org

1, 2, 7, 34, 173, 1066, 7147, 51962, 412729, 3478258, 31220111, 296409202, 2953487077, 30870965594, 336796018483, 3824230997386, 45114077004017, 551338045973602, 6968344940992279, 90931562913957698, 1222939213021853341, 16929504703420184842, 240909000856701880187
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2024

Keywords

Crossrefs

Cf. A018191, A377964.
Cf. A361278, A377965.

Programs

  • PARI
    a(n, s=1, t=2) = n!*sum(k=0, n, binomial(t*k+s, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} binomial(2*k+1,n-k) / k!.
a(n) = a(n-1) + (4*n-3)*a(n-2) + 3*(n-2)*n*a(n-3) for n > 2.

A377956 a(n) = n! * Sum_{k=0..n} binomial(k+4,n-k) / k!.

Original entry on oeis.org

1, 5, 23, 103, 473, 2261, 11215, 57863, 309713, 1715653, 9831911, 58058375, 353546473, 2210900693, 14215319903, 93610866151, 632159025185, 4362925851653, 30809311250743, 221958273142823, 1632956199823481, 12238229941781845, 93509510960341103, 726913018468699463
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2024

Keywords

Crossrefs

Cf. A018191, A047974, A377954, A377955.

Programs

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

Formula

E.g.f.: (1 + x)^4 * exp(x + x^2).
a(n) = -(n-6)*a(n-1) + 3*(n-1)*a(n-2) + 2*(n-1)*(n-2)*a(n-3) for n > 2.

A377958 Expansion of e.g.f. exp(x - x^2)/(1 - x).

Original entry on oeis.org

1, 2, 3, 4, 17, 126, 787, 5048, 39489, 361882, 3641411, 39948492, 478777873, 6226077014, 87182747667, 1307703873856, 20922694556417, 355686434950578, 6402375749061379, 121645136562423572, 2432901971620591761, 51090940751194252462, 1124000727777806326163
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2024

Keywords

Crossrefs

Cf. A293604, A377959, A377960.
Cf. A018191.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} binomial(n-2*k,n-k) / k!.
a(n) = (n+1)*a(n-1) - 3*(n-1)*a(n-2) + 2*(n-1)*(n-2)*a(n-3) for n > 2.

A380617 Triangle read by rows: T(n,k) is the number of achiral combinatorial maps with n edges and k vertices, 1 <= k <= n + 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 8, 5, 2, 16, 28, 26, 12, 3, 53, 121, 128, 82, 28, 6, 206, 528, 686, 505, 239, 68, 10, 817, 2516, 3638, 3192, 1802, 686, 157, 20, 3620, 12302, 20250, 19976, 13268, 6078, 1876, 372, 35, 16361, 63643, 114669, 126876, 95422, 50954, 19346, 5100, 845, 70
Offset: 0

Views

Author

Andrew Howroyd, Jan 28 2025

Keywords

Comments

By duality, also the number of achiral combinatorial maps with n edges and k faces.

Examples

			Triangle begins:
n\k |    1      2      3      4      5     6     7    8   9
----+-------------------------------------------------------
  0 |    1;
  1 |    1,     1;
  2 |    2,     2,     1;
  3 |    5,     8,     5,     2;
  4 |   16,    28,    26,    12,     3;
  5 |   53,   121,   128,    82,    28,    6;
  6 |  206,   528,   686,   505,   239,   68,   10;
  7 |  817,  2516,  3638,  3192,  1802,  686,  157,  20;
  8 | 3620, 12302, 20250, 19976, 13268, 6078, 1876, 372, 35;
  ...

Crossrefs

Row sums are A170947.
Main diagonal is A001405(n-1).
Column 1 is A018191.
Cf. A379431 (planar), A380615 (sensed), A380616 (unsensed).

A054938 Number of chiral chord diagrams on n nodes.

Original entry on oeis.org

0, 0, 0, 1, 26, 348, 4466, 61726, 949795, 16331482, 312298796, 6587217199, 152030203190, 3811719561156, 103171205826822, 2998417379370294, 93127344062857976, 3078376281077418971, 107905190923235526392
Offset: 1

Views

Author

N. J. A. Sloane, May 24 2000

Keywords

Links

Formula

a(n) = A054499(n)-A018191(n). [Liskovets, referring to offset 1 in A054499]. - R. J. Mathar, Oct 01 2011
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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