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.

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A380014 Expansion of e.g.f. 1/sqrt(exp(-2*x) - 2*x).

Original entry on oeis.org

1, 2, 10, 88, 1084, 17176, 332824, 7623904, 201540112, 6038820640, 202246657696, 7486877795200, 303561658686400, 13378863292503424, 636833910410881408, 32559375816074384896, 1779494669204225605888, 103532173699456380625408, 6388705590982575700625920
Offset: 0

Views

Author

Seiichi Manyama, Jan 09 2025

Keywords

Crossrefs

Cf. A072597, A380016, A380018.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (-2)^k * (2*k+1)^(n-k) * binomial(-1/2,k)/(n-k)!.
a(n) ~ 2^(n + 1/2) * n^n / (sqrt(1 + LambertW(1)) * LambertW(1)^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Jan 23 2025

A380016 Expansion of e.g.f. 1/(exp(-3*x) - 3*x)^(1/3).

Original entry on oeis.org

1, 2, 13, 161, 2833, 64841, 1827685, 61192181, 2372620801, 104549934977, 5160225776101, 281994042839477, 16902276273364465, 1102519010117525105, 77749077431938305541, 5894145002422856684501, 478015727336387513545345, 41295912476641866286397825, 3786025873450493919700627525
Offset: 0

Views

Author

Seiichi Manyama, Jan 09 2025

Keywords

Crossrefs

Cf. A072597, A380014, A380018.

Programs

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

Formula

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

A368236 Expansion of e.g.f. 1/(exp(-x) - 2*x).

Original entry on oeis.org

1, 3, 17, 145, 1649, 23441, 399865, 7957881, 180997857, 4631289697, 131670338921, 4117813225769, 140486274499345, 5192341564319313, 206669931188282073, 8813624820931402201, 400922608851086766017, 19377398675442025382081, 991639882680576890150089
Offset: 0

Views

Author

Seiichi Manyama, Dec 18 2023

Keywords

Crossrefs

Cf. A072597, A368237.
Cf. A336947, A368233.

Programs

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

Formula

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

A375394 Expansion of e.g.f. 1 / (exp(-x^2/2) - x).

Original entry on oeis.org

1, 1, 3, 12, 63, 420, 3345, 31080, 330225, 3946320, 52401195, 765404640, 12196214415, 210533843520, 3913845680745, 77955813936000, 1656235524168225, 37387344753158400, 893615568162592275, 22545399132243187200, 598744483093370188575, 16696076277239091532800
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2024

Keywords

Crossrefs

Cf. A072597, A375395.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} ((n-2*k+1)/2)^k/k!.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / ((1 + LambertW(1)) * exp(n) * LambertW(1)^((n+1)/2)). - Vaclav Kotesovec, Aug 21 2024

A375604 Expansion of e.g.f. 1 / (exp(-x^2) - x).

Original entry on oeis.org

1, 1, 4, 18, 108, 840, 7680, 82320, 1009680, 13910400, 213071040, 3589850880, 65975152320, 1313624632320, 28166959941120, 647099547494400, 15857424488505600, 412878579034521600, 11382450106662835200, 331230511848421785600, 10146149192841050188800
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2024

Keywords

Crossrefs

Cf. A072597, A375607.
Cf. A375394, A375608.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(Exp[-x^2]-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 01 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(exp(-x^2)-x)))
  • PARI
    a(n) = n!*sum(k=0, n\2, (n-2*k+1)^k/k!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k+1)^k/k!.
a(n) ~ sqrt(Pi) * 2^(n/2 + 1) * n^(n + 1/2) / ((1 + LambertW(2)) * exp(n) * LambertW(2)^((n+1)/2)). - Vaclav Kotesovec, Aug 21 2024

A379942 Expansion of e.g.f. 1/( exp(-x) - x )^3.

Original entry on oeis.org

1, 6, 45, 411, 4449, 55803, 796581, 12757503, 226588257, 4420898595, 94001021589, 2163619250895, 53598352999905, 1421924243354787, 40221778417553637, 1208471542554184767, 38434396264371831873, 1289995362325669726659, 45567027291743788320405
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Cf. A072597, A379933, A379943.
Cf. A377530.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(exp(-x)-x)^3))
  • PARI
    a(n) = n!*sum(k=0, n, (k+3)^(n-k)*binomial(k+2, 2)/(n-k)!);

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A072597.
a(n) = n! * Sum_{k=0..n} (k+3)^(n-k) * binomial(k+2,2)/(n-k)!.

A336949 a(n) = n! * [x^n] 1 / (exp(-n*x) - x).

Original entry on oeis.org

1, 2, 14, 195, 4440, 147745, 6698448, 394852577, 29250137472, 2652483234033, 288363456748800, 36952298766628465, 5504130616452258816, 941845623036360908489, 183298110723156455921664, 40221612394630225987208625, 9876429434585097671993032704
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 08 2020

Keywords

Crossrefs

Cf. A063170, A072597, A235328, A336947, A336948.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[1/(Exp[-n x] - x), {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[n! Sum[(n (n - k + 1))^k/k!, {k, 0, n}], {n, 1, 16}]]
  • PARI
    a(n)={n!*polcoef(1/(exp(-n*x + O(x*x^n)) - x), n)} \\ Andrew Howroyd, Aug 08 2020

Formula

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

A375607 Expansion of e.g.f. 1 / (exp(-x^3) - x).

Original entry on oeis.org

1, 1, 2, 12, 72, 480, 3960, 40320, 463680, 5866560, 82857600, 1297296000, 22133865600, 407869862400, 8096683795200, 172405968134400, 3915525770956800, 94443904345190400, 2412049832704512000, 65035187612185190400, 1845812342328514560000
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2024

Keywords

Crossrefs

Cf. A072597, A375604.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(exp(-x^3)-x)))
  • PARI
    a(n) = n!*sum(k=0, n\3, (n-3*k+1)^k/k!);

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k+1)^k/k!.
a(n) ~ sqrt(2*Pi) * 3^((n+1)/3) * n^(n + 1/2) / ((1 + LambertW(3)) * exp(n) * LambertW(3)^((n+1)/3)). - Vaclav Kotesovec, Aug 21 2024

A377740 E.g.f. satisfies A(x) = exp(x) * (1 + x * A(x))^2.

Original entry on oeis.org

1, 3, 19, 199, 2957, 57341, 1377175, 39531927, 1321803705, 50491876825, 2170432191491, 103726081148339, 5456983990544773, 313449393386822421, 19521567325327386831, 1310428405901227674511, 94325931842372734994417, 7248016420075574268626225, 592190617414334419733622139
Offset: 0

Views

Author

Seiichi Manyama, Nov 05 2024

Keywords

Crossrefs

Cf. A072597, A377741.
Cf. A295238, A377574.

Programs

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

Formula

E.g.f.: 4*exp(x)/(1 + sqrt(1 - 4*x*exp(x)))^2.
a(n) = n! * Sum_{k=0..n} (k+1)^(n-k-1) * binomial(2*k+2,k)/(n-k)!.
a(n) = A295238(n+1)/(n+1).

A380018 Expansion of e.g.f. 1/(exp(-4*x) - 4*x)^(1/4).

Original entry on oeis.org

1, 2, 16, 256, 5856, 175296, 6486016, 285756416, 14606007296, 849615763456, 55415153442816, 4005309938466816, 317750919017168896, 27449350209163821056, 2564871898004949303296, 257753802183061443444736, 27720748513211258671988736, 3176821722223524679312736256
Offset: 0

Views

Author

Seiichi Manyama, Jan 09 2025

Keywords

Crossrefs

Cf. A072597, A380014, A380016.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (-4)^k*(4*k+1)^(n-k)*binomial(-1/4, k)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} (-4)^k * (4*k+1)^(n-k) * binomial(-1/4,k)/(n-k)!.
Previous Showing 11-20 of 27 results. Next

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