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 1-10 of 11 results. Next

A349601 E.g.f. satisfies: A(x) * log(A(x)) = exp(x*A(x)^2) - 1.

Original entry on oeis.org

1, 1, 4, 32, 391, 6462, 134974, 3412187, 101323674, 3457536144, 133333945461, 5734792007584, 272197255745078, 14133109419794601, 796883164532719216, 48489515568651113516, 3167153388603620859695, 221021628292403019655418
Offset: 0

Views

Author

Seiichi Manyama, Nov 22 2021

Keywords

Links

Crossrefs

Cf. A216135, A349600, A349602.
Cf. A349583, A349588.

Programs

  • Mathematica
    Table[Sum[(2*n - k + 1)^(k-1) * StirlingS2[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 25 2021 *)
  • PARI
    a(n) = sum(k=0, n, (2*n-k+1)^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (2*n-k+1)^(k-1) * Stirling2(n,k).
a(n) ~ s * n^(n-1) / (sqrt(2*(1 + 2*r*s^2) - 2/(1 + log(s))) * exp(n) * r^n), where r = 0.2229533052706631261980294005821031136702825459439... and s = 1.759796045489338472919926226485178994146849909897... are roots of the system of equations exp(r*s^2) = 1 + s*log(s), 2*exp(r*s^2)*r*s = 1 + log(s). - Vaclav Kotesovec, Nov 25 2021

A349587 E.g.f. satisfies: A(x)^A(x) = 1 + x*A(x).

Original entry on oeis.org

1, 1, 0, -3, 4, 60, -294, -2800, 34504, 197568, -6087360, -9146808, 1488986808, -5886157992, -469973309064, 5492298353880, 177826238321856, -4277426240130048, -72353540601814464, 3537861051231290880, 22847222673714931200, -3226666120379253611136
Offset: 0

Views

Author

Seiichi Manyama, Nov 22 2021

Keywords

Links

Crossrefs

Cf. A141209, A349588, A349589.
Cf. A120980, A162655, A162656.

Programs

  • Mathematica
    a[n_] := Sum[(n - k + 1)^(k - 1)*StirlingS1[n, k], {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, Nov 23 2021 *)
  • PARI
    a(n) = sum(k=0, n, (n-k+1)^(k-1)*stirling(n, k, 1));

Formula

a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * Stirling1(n,k).

A349589 E.g.f. satisfies: A(x) * log(A(x)) = 1 - exp(-x*A(x)).

Original entry on oeis.org

1, 1, 0, -4, -3, 87, 230, -4583, -27216, 434928, 4871719, -62913079, -1240374960, 12230778601, 426135019232, -2759957884648, -189393687667107, 479371576805751, 105233549909615798, 233116575802412969, -71022416772836562008, -574100485456271792020
Offset: 0

Views

Author

Seiichi Manyama, Nov 22 2021

Keywords

Links

Crossrefs

Cf. A141209, A349587, A349588.
Cf. A349558.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(n - k)*(n - k + 1)^(k - 1)*StirlingS2[n, k], {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, Nov 23 2021 *)
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*(n-k+1)^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * (n-k+1)^(k-1) * Stirling2(n,k).

A357424 E.g.f. satisfies A(x) * exp(A(x)) = exp(x * exp(A(x))) - 1.

Original entry on oeis.org

0, 1, 1, 4, 21, 156, 1470, 16843, 227367, 3533974, 62163477, 1220852524, 26480355110, 628693388909, 16216901961481, 451609382251836, 13504072800481613, 431544662700594212, 14677503631085378170, 529370720888418692643, 20180856622352239827687
Offset: 0

Views

Author

Seiichi Manyama, Sep 27 2022

Keywords

Links

Crossrefs

Cf. A052888, A357346, A357347, A357348.
Cf. A349588.

Programs

  • Mathematica
    Join[{0,1},Table[Sum[(n-k)^(k-1) * StirlingS2[n,k], {k,1,n}], {n,2,20}]] (* Vaclav Kotesovec, Nov 14 2022 *)
  • PARI
    a(n) = sum(k=1, n, (n-k)^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=1..n} (n-k)^(k-1) * Stirling2(n,k).
a(n) ~ n^(n-1) * (1 + exp(s)*s)^(n + 1/2) / (sqrt(exp(s)*(1 + s + s^2) - 1) * exp(n) * (1 + s)^(n - 1/2)), where s = 1.104072744884035178291292242554731... is the root of the equation 1 + s = (exp(-s) + s) * log(1 + exp(s)*s). - Vaclav Kotesovec, Nov 14 2022
E.g.f.: Series_Reversion( exp(-x) * log(1 + x * exp(x)) ). - Seiichi Manyama, Sep 09 2024

A355763 E.g.f. satisfies A(x)^2 * log(A(x)) = exp(x*A(x)) - 1.

Original entry on oeis.org

1, 1, 0, 5, -13, 207, -1791, 28849, -438600, 8619291, -181134313, 4381744589, -115439041983, 3356162869607, -105668550658100, 3600058076291465, -131618721053773713, 5146452228945999699, -214171122214841864975, 9454288479242533668837
Offset: 0

Views

Author

Seiichi Manyama, Jul 16 2022

Keywords

Crossrefs

Cf. A030019, A349588.
Cf. A349654, A355762.

Programs

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

Formula

a(n) = Sum_{k=0..n} (n-2*k+1)^(k-1) * Stirling2(n,k).

A357088 E.g.f. satisfies A(x) * log(A(x)) = (exp(x*A(x)) - 1)^2 / 2.

Original entry on oeis.org

1, 0, 1, 3, 16, 135, 1246, 14238, 192613, 2948025, 51071236, 985911003, 20952667660, 486857940660, 12275673296251, 333786662478363, 9737819506544272, 303399477464036175, 10054949172135522106, 353197317869395005258, 13108298181041284002769
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2022

Keywords

Crossrefs

Cf. A349588, A357089.

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/2)} (2*k)! * (n-k+1)^(k-1) * Stirling2(n,2*k)/(2^k * k!).

A357089 E.g.f. satisfies A(x) * log(A(x)) = (exp(x*A(x)) - 1)^3 / 6.

Original entry on oeis.org

1, 0, 0, 1, 6, 25, 140, 1561, 19586, 228425, 2870160, 44172601, 780614846, 14499946825, 284310704860, 6089231941521, 142225796401786, 3537029819020905, 92766573133851240, 2577870903366020521, 75999605064376599606, 2362944241092314079145
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2022

Keywords

Crossrefs

Cf. A349588, A357088.

Programs

  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*(n-k+1)^(k-1)*stirling(n, 3*k, 2)/(6^k*k!));

Formula

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * (n-k+1)^(k-1) * Stirling2(n,3*k)/(6^k * k!).

A356882 E.g.f. satisfies: A(x) * log(A(x)) = x * (exp(x*A(x)) - 1).

Original entry on oeis.org

1, 0, 2, 3, 16, 125, 756, 7567, 85968, 994905, 14373460, 225366251, 3800667960, 72169966453, 1469546796732, 32150706096615, 760806334538656, 19142440567996721, 512272692571487652, 14560087915617858883, 436598686303562722440, 13796641165956117509901
Offset: 0

Views

Author

Seiichi Manyama, Sep 02 2022

Keywords

Crossrefs

Cf. A349560, A356785, A356788, A356789, A356883.
Cf. A349588.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.

A357011 E.g.f. satisfies A(x) * log(A(x)) = exp(x * A(x)^3) - 1.

Original entry on oeis.org

1, 1, 6, 74, 1407, 36357, 1190476, 47254783, 2205546706, 118378505742, 7184030384361, 486440226752911, 36358328607088010, 2973464028723984551, 264119772408892921774, 25321946948812001539166, 2606224408648404660237647, 286624141573198517220290837
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2022

Keywords

Links

Crossrefs

Cf. A349583, A349588, A349601.
Cf. A216136, A356960, A356973.

Programs

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

Formula

a(n) = Sum_{k=0..n} (3*n-k+1)^(k-1) * Stirling2(n,k).

A357086 E.g.f. satisfies A(x) * log(A(x)) = (exp(x*A(x)) - 1)^2.

Original entry on oeis.org

1, 0, 2, 6, 50, 510, 5882, 88326, 1502258, 29368590, 650366762, 15974149686, 433095937826, 12829712583870, 412295632858202, 14292175302568806, 531485147656990994, 21107739762958541550, 891673745283286886282, 39923664347178352362006
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2022

Keywords

Crossrefs

Cf. A349588, A357087.
Cf. A357084, A357088.

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/2)} (2*k)! * (n-k+1)^(k-1) * Stirling2(n,2*k)/k!.
Showing 1-10 of 11 results. Next

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