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.

A368265 Expansion of e.g.f. exp(2*x) / (1 - x*exp(x)).

Original entry on oeis.org

1, 3, 12, 65, 460, 4057, 42922, 529769, 7472808, 118586033, 2090936014, 40554647377, 858082563532, 19668880007129, 485528656965762, 12841428220413593, 362276791422785488, 10859170086870710497, 344648459867067117334, 11546148650974694099201
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2023

Keywords

Crossrefs

Cf. A006153, A072597, A092148, A368266.
Cf. A368267, A368271.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (n-k+2)^k / k!.
a(n) ~ n! / ((1 + LambertW(1)) * LambertW(1)^(n+2)). - Vaclav Kotesovec, Dec 29 2023

A368266 Expansion of e.g.f. exp(-2*x) / (1 - x*exp(x)).

Original entry on oeis.org

1, -1, 4, 1, 60, 393, 4498, 54689, 773384, 12268369, 216328614, 4195769433, 88776964444, 2034936319817, 50232646818890, 1328570248040497, 37481046375146640, 1123486426007081505, 35657224567565828302, 1194561018775753556777, 42125545306641497600036
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2023

Keywords

Crossrefs

Cf. A006153, A072597, A092148, A368265.

Programs

  • Mathematica
    CoefficientList[Series[Exp[-2*x]/ (1 - x*Exp[x]),{x,0,20}],x]Range[0,20]! (* Stefano Spezia, Feb 21 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n, (n-k-2)^k/k!);

Formula

a(n) = n! * Sum_{k=0..n} (n-k-2)^k / k!.

A368294 Expansion of e.g.f. exp(-x) / (1 - x*exp(2*x)).

Original entry on oeis.org

1, 0, 5, 26, 257, 3004, 42277, 694182, 13027233, 275027480, 6451456421, 166468606930, 4685926374241, 142896198878484, 4692784128725253, 165121528707826334, 6197343209916830657, 247136182723119609520, 10434957969449887989445, 465078400154112693602826
Offset: 0

Views

Author

Seiichi Manyama, Dec 20 2023

Keywords

Crossrefs

Cf. A092148, A368295.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (2*(n-k)-1)^k/k!);

Formula

a(n) = n! * Sum_{k=0..n} (2*(n-k)-1)^k / k!.

A368295 Expansion of e.g.f. exp(-x) / (1 - x*exp(3*x)).

Original entry on oeis.org

1, 0, 7, 47, 549, 7859, 134743, 2694537, 61595593, 1584039383, 45262146411, 1422647401493, 48780718512397, 1812012517295811, 72486789793754911, 3106846273709042609, 142039582571511632529, 6899664100487740573871, 354870658975988082859219
Offset: 0

Views

Author

Seiichi Manyama, Dec 20 2023

Keywords

Crossrefs

Cf. A092148, A368294.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (3*(n-k)-1)^k/k!);

Formula

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

A379990 Expansion of e.g.f. exp(-2*x)/(exp(-x) - x)^3.

Original entry on oeis.org

1, 4, 25, 205, 2065, 24601, 337837, 5249581, 91006657, 1740663937, 36402220141, 826159146253, 20220201899377, 530828186303377, 14878044338021677, 443397290411503021, 14000282854007503105, 466866129420834410881, 16395362179348570608205, 604794784980600986425645
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Cf. A377530, A379942, A379991.
Cf. A092148, A377529.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k) * binomial(k+2,2)/(n-k)!.
a(n) ~ n! * n^2 / (2 * (LambertW(1) + 1)^3 * LambertW(1)^(n+1)). - Vaclav Kotesovec, Jan 08 2025

A379997 Expansion of e.g.f. 1/(exp(x) - x*exp(2*x))^2.

Original entry on oeis.org

1, 0, 6, 22, 224, 2138, 25732, 351846, 5458224, 94441042, 1803255404, 37652268014, 853321021192, 20858236815258, 546941712302052, 15313467390967222, 455933682027961184, 14383416438784605602, 479254037890010238172, 16817855455956128823486, 619953003446894086537656
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Cf. A358738, A377529, A379933, A379992.
Cf. A092148.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A092148.
a(n) = n! * Sum_{k=0..n} (k+1) * (k-2)^(n-k)/(n-k)!.
Showing 1-6 of 6 results.

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

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