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.

A052851 Expansion of e.g.f. 1/2 - (1/2)*(1+4*log(1-x))^(1/2).

Original entry on oeis.org

0, 1, 3, 20, 220, 3424, 69008, 1706256, 49956240, 1689497376, 64799254752, 2778906776832, 131756614920192, 6843405231815424, 386414606189283072, 23567401521343170048, 1543994621969805135360, 108137637714495023354880, 8062825821198926369725440
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

Previous name was: A simple grammar.

Links

Crossrefs

Cf. A048287, A052886, A087138.
Cf. A371314, A371315.
Cf. A371327, A376041, A376042.

Programs

  • Maple
    spec := [S,{B=Cycle(Z),S=Prod(B,C),C=Sequence(S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    CoefficientList[Series[1/2-1/2*(1+4*Log[1-x])^(1/2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)
  • Maxima
    a(n):=sum(stirling1(n,k)*k!*binomial(2*k-2,k-1)/k*(-1)^(n+k), k,1,n); /* Vladimir Kruchinin, May 12 2012 */

Formula

E.g.f.: 1/2 - (1/2)*(1-4*log(-1/(-1+x)))^(1/2).
a(n) = Sum_{k=1..n} Stirling1(n,k)*k!*C(2*k-2,k-1)/k*(-1)^(n+k). - Vladimir Kruchinin, May 12 2012
a(n) ~ n^(n-1)/(sqrt(2)*exp(3*n/4)*(exp(1/4)-1)^(n-1/2)). - Vaclav Kotesovec, Sep 30 2013
From Seiichi Manyama, Sep 09 2024: (Start)
E.g.f. satisfies A(x) = (-log(1 - x)) / (1 - A(x)).
E.g.f.: Series_Reversion( 1 - exp(-x * (1 - x)) ). (End)

Extensions

New name using e.g.f., Vaclav Kotesovec, Sep 30 2013

A371327 E.g.f. satisfies A(x) = -log(1 - x/(1 - A(x)))/(1 - A(x)).

Original entry on oeis.org

0, 1, 5, 59, 1128, 29954, 1019282, 42318296, 2074276320, 117237652008, 7506386360232, 536983774338120, 42447806791009056, 3674351246886880416, 345667310491536157056, 35116581800947400780928, 3831441153568328284066560, 446832269484565155280539264
Offset: 0

Views

Author

Seiichi Manyama, Mar 19 2024

Keywords

Links

Crossrefs

Cf. A052842, A371328.
Cf. A052851, A376041, A376042.

Programs

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

Formula

a(n) = Sum_{k=1..n} (n+2*k-2)!/(n+k-1)! * |Stirling1(n,k)|.
E.g.f.: Series_Reversion( (1 - x) * (1 - exp(-x * (1 - x))) ). - Seiichi Manyama, Sep 08 2024

A376042 E.g.f. satisfies A(x) = (-log(1 - x / (1 - A(x))^2)) / (1 - A(x)).

Original entry on oeis.org

0, 1, 7, 116, 3092, 114034, 5378396, 309151968, 20964872624, 1638608258904, 145038615271512, 14340344355439200, 1566483453363376896, 187355848936261332144, 24351019737412176648576, 3417500066845923960657408, 515071814323666902383222784
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2024

Keywords

Links

Crossrefs

Cf. A052851, A371327, A376041.

Programs

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

Formula

a(n) = Sum_{k=1..n} (2*n+2*k-2)!/(2*n+k-1)! * |Stirling1(n,k)|.
E.g.f.: Series_Reversion( (1 - x)^2 * (1 - exp(-x * (1 - x))) ).

A376038 E.g.f. satisfies A(x) = (-log(1 - x / (1 - A(x))^3)) * (1 - A(x))^2.

Original entry on oeis.org

0, 1, 3, 29, 466, 10444, 300296, 10539738, 436831368, 20879226240, 1130604893016, 68406042884376, 4573574072262240, 334855813955693952, 26645202689658107712, 2289609993045578793120, 211302073839493597484160, 20844012997702684830894336
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2024

Keywords

Links

Crossrefs

Cf. A376039, A376040, A376041.

Programs

  • PARI
    a(n) = sum(k=1, n, (3*n-k-2)!/(3*n-2*k-1)!*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=1..n} (3*n-k-2)!/(3*n-2*k-1)! * |Stirling1(n,k)|.
E.g.f.: Series_Reversion( (1 - x)^3 * (1 - exp(-x / (1 - x)^2)) ).

A376039 E.g.f. satisfies A(x) = (-log(1 - x / (1 - A(x))^3)) * (1 - A(x)).

Original entry on oeis.org

0, 1, 5, 65, 1376, 40454, 1523464, 69979734, 3794288280, 237186275520, 16794542216088, 1328558461234080, 116126748206895216, 11114654375545182864, 1156103394150386866560, 129855826037621953356864, 15664344145032570448561920, 2019701492029961287845196032
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2024

Keywords

Links

Crossrefs

Cf. A376038, A376040, A376041.

Programs

  • PARI
    a(n) = sum(k=1, n, (3*n-2)!/(3*n-k-1)!*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=1..n} (3*n-2)!/(3*n-k-1)! * |Stirling1(n,k)|.
E.g.f.: Series_Reversion( (1 - x)^3 * (1 - exp(-x / (1 - x))) ).

A376040 E.g.f. satisfies A(x) = -log(1 - x / (1 - A(x))^3).

Original entry on oeis.org

0, 1, 7, 119, 3258, 123414, 5978082, 352880562, 24573720672, 1972239280488, 179250831525720, 18197871488362752, 2041093578923498448, 250654006995798120480, 33449544716000374458000, 4819960747934844400104480, 745867334512204468875843840
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2024

Keywords

Links

Crossrefs

Cf. A376038, A376039, A376041.
Cf. A371370.

Programs

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

Formula

E.g.f.: Series_Reversion( (1 - x)^3 * (1 - exp(-x)) ).
a(n) = Sum_{k=1..n} (3*n+k-2)!/(3*n-1)! * |Stirling1(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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