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 1-5 of 5 results.

A094418 Generalized ordered Bell numbers Bo(5,n).

Original entry on oeis.org

1, 5, 55, 905, 19855, 544505, 17919055, 687978905, 30187495855, 1490155456505, 81732269223055, 4931150091426905, 324557348772511855, 23141780973332248505, 1776997406800302687055, 146197529083891406394905, 12829862285488250150167855, 1196280147496701351115120505
Offset: 0

Views

Author

Ralf Stephan, May 02 2004

Keywords

Comments

Fifth row of array A094416, which has more information.

Links

Crossrefs

Cf. A094416, A094417, A094419, A094422, A131689.
Cf. A346984, A365568, A365569, A365570.

Programs

  • Magma
    A094416:= func< n,k | (&+[Factorial(j)*n^j*StirlingSecond(k,j): j in [0..k]]) >;
A094418:= func< k | A094416(5,k) >;
[A094418(n): n in [0..30]]; // G. C. Greubel, Jan 12 2024
  • Mathematica
    t = 30; Range[0, t]! CoefficientList[Series[1/(6 - 5 Exp[x]), {x, 0, t}], x] (* Vincenzo Librandi, Mar 16 2014 *)
  • PARI
    my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(6 - 5*exp(x)))) \\ Joerg Arndt, Jan 15 2024
  • SageMath
    def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
def A094418(k): return A094416(5,k)
[A094418(n) for n in range(31)] # G. C. Greubel, Jan 12 2024

Formula

E.g.f.: 1/(6 - 5*exp(x)).
a(n) = Sum_{k=0..n} A131689(n,k) * 5^k. - Philippe Deléham, Nov 03 2008
a(n) ~ n! / (6*(log(6/5))^(n+1)). - Vaclav Kotesovec, Mar 14 2014
a(0) = 1; a(n) = 5 * Sum_{k=1..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020
a(0) = 1; a(n) = 5 * a(n-1) - 6 * Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023
From Seiichi Manyama, Jun 01 2025: (Start)
a(n) = (-1)^(n+1)/6 * Li_{-n}(6/5), where Li_{n}(x) is the polylogarithm function.
a(n) = (1/6) * Sum_{k>=0} k^n * (5/6)^k.
a(n) = (5/6) * Sum_{k=0..n} 6^k * (-1)^(n-k) * A131689(n,k) for n > 0. (End)

A365568 Expansion of e.g.f. 1 / (6 - 5 * exp(x))^(2/5).

Original entry on oeis.org

1, 2, 16, 212, 3964, 95804, 2840140, 99760124, 4050900268, 186700658972, 9628444876108, 549349531209404, 34355463031007596, 2336935606239856988, 171779270567736231052, 13568895740353218626300, 1146225546710339427328684, 103113032296428007394503580
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2023

Keywords

Crossrefs

Cf. A094418, A346984, A365569, A365570.

Programs

  • Mathematica
    a[n_] := Sum[Product[5*j + 2, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Sep 11 2023 *)
  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 5*j+2)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (5*j+2)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (5 - 3*k/n) * binomial(n,k) * a(n-k).
a(n) ~ sqrt(Pi) * 2^(1/10) * n^(n - 1/10) / (3^(2/5) * Gamma(2/5) * exp(n) * log(6/5)^(n + 2/5)). - Vaclav Kotesovec, Nov 11 2023
a(0) = 1; a(n) = 2*a(n-1) - 6*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

A365569 Expansion of e.g.f. 1 / (6 - 5 * exp(x))^(3/5).

Original entry on oeis.org

1, 3, 27, 387, 7659, 193491, 5948091, 215446563, 8984708235, 423944899443, 22328393101659, 1298429924941251, 82625791930962219, 5711012035686681363, 426058604580805219323, 34121803137713388036963, 2919847869159667841599947, 265868538017899566748612275
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2023

Keywords

Crossrefs

Cf. A094418, A346984, A365568, A365570.

Programs

  • Mathematica
    a[n_] := Sum[Product[5*j + 3, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Sep 11 2023 *)
With[{nn=20},CoefficientList[Series[1/(6-5*Exp[x])^(3/5),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 03 2024 *)
  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 5*j+3)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (5*j+3)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (5 - 2*k/n) * binomial(n,k) * a(n-k).
a(n) ~ sqrt(2*Pi) * n^(n + 1/10) / (6^(3/5) * Gamma(3/5) * exp(n) * log(6/5)^(n + 3/5)). - Vaclav Kotesovec, Nov 11 2023
a(0) = 1; a(n) = 3*a(n-1) - 6*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

A365587 Expansion of e.g.f. 1 / (1 + 5 * log(1-x))^(4/5).

Original entry on oeis.org

1, 4, 40, 620, 13020, 345120, 11049960, 414711720, 17851113720, 866838536640, 46873882199520, 2793214943693280, 181854240448514400, 12842833148474299200, 977822088984613771200, 79842750450344086867200, 6959878576257689846265600
Offset: 0

Views

Author

Seiichi Manyama, Sep 10 2023

Keywords

Crossrefs

Cf. A346987, A365585, A365586, A365588.
Cf. A365570.

Programs

  • Mathematica
    a[n_] := Sum[Product[5*j + 4, {j, 0, k - 1}] * Abs[StirlingS1[n, k]], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 13 2023 *)
  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 5*j+4)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (5*j+4)) * |Stirling1(n,k)|.
a(0) = 1; a(n) = Sum_{k=1..n} (5 - k/n) * (k-1)! * binomial(n,k) * a(n-k).

A367376 Expansion of the e.g.f. (exp(x) / (5 - 4*exp(x)))^(4/5).

Original entry on oeis.org

1, 4, 32, 400, 6800, 146128, 3795728, 115616848, 4040024720, 159282704848, 6993908053520, 338443123424080, 17894609985867152, 1026351961130219728, 63466858180767590672, 4209071260503851502160, 298006515851074633361552, 22434758711582422326267856
Offset: 0

Views

Author

Seiichi Manyama, Nov 15 2023

Keywords

Crossrefs

Cf. A136729, A201365, A367374, A367375.
Cf. A365570, A365587, A365603.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*prod(j=0, k-1, 5*j+4)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * (Product_{j=0..k-1} (5*j+4)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * (k/n - 5) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 4*a(n-1) + 4*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k).
Showing 1-5 of 5 results.

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