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.

A052830 A simple grammar: sequences of rooted cycles.

Original entry on oeis.org

1, 0, 2, 3, 32, 150, 1524, 12600, 147328, 1705536, 23681520, 345605040, 5654922624, 98624766240, 1870594556544, 37794037488480, 817362198512640, 18742996919324160, 455648694329309184, 11683777530785978880, 315505598702787118080, 8943481464393674096640
Offset: 0

Views

Author

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

Keywords

Comments

Asymptotic behavior (formula 3.2.) in the INRIA reference is wrong! - Vaclav Kotesovec, Jun 03 2019

Links

Crossrefs

Cf. A353880, A353881, A353882.
Cf. A052848, A066166.

Programs

  • Maple
    spec := [S,{B=Prod(C,Z),C=Cycle(Z),S=Sequence(B)},labeled]: seq(combstruct[count](spec, size=n), n=0..20);
  • Mathematica
    CoefficientList[Series[1/(1+x*Log[1-x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)
  • Maxima
    a(n):=(-1)^(n)*n!*sum((k!*stirling1(n-k,k))/(n-k)!,k,0,n/2); /* Vladimir Kruchinin, Nov 16 2011 */
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!*sum(j=2, i, 1/(j-1)*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, May 04 2022
  • PARI
    a(n) = n!*sum(k=0, n\2, k!*abs(stirling(n-k, k, 1))/(n-k)!); \\ Seiichi Manyama, May 04 2022

Formula

E.g.f.: 1/(1-x*log(1/(1-x))).
a(n) = (-1)^n*n!*Sum_{k=0..floor(n/2)} k!*Stirling1(n-k,k)/(n-k)!. - Vladimir Kruchinin, Nov 16 2011
a(n) ~ n! * r^(n+1)/(r+1/(r-1)), where r = 1.349976485401125... is the root of the equation (r-1)*exp(r) = r. - Vaclav Kotesovec, Sep 30 2013
a(0) = 1; a(n) = n! * Sum_{k=2..n} 1/(k-1) * a(n-k)/(n-k)!. - Seiichi Manyama, May 04 2022

Extensions

More terms from Alois P. Heinz, Mar 16 2016

A353881 Expansion of e.g.f. 1/(1 + (x * log(1-x))^3 / 36).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 20, 210, 1960, 18900, 194880, 2166780, 26356880, 349806600, 5029088064, 77748751080, 1284349422720, 22551300670080, 419191223208384, 8222848137607680, 169760091173740800, 3679746265902067200, 83564915096633308800, 1984162781781147770880
Offset: 0

Views

Author

Seiichi Manyama, May 09 2022

Keywords

Crossrefs

Cf. A052830, A353880, A353882.
Cf. A346922, A353884.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+(x*log(1-x))^3/36)))
  • PARI
    a(n) = n!*sum(k=0, n\6, (3*k)!*abs(stirling(n-3*k, 3*k, 1))/(36^k*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/6)} (3*k)! * |Stirling1(n-3*k,3*k)|/(36^k * (n-3*k)!).

A353882 Expansion of e.g.f. 1/(1 - (x * log(1-x))^4 / 576).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 70, 1260, 17850, 242550, 3350655, 48108060, 724403680, 11478967500, 191632761320, 3369643717440, 62346624827760, 1212116258480400, 24721764604046280, 528066880710319440, 11793526736005503720, 274937000436908714520
Offset: 0

Views

Author

Seiichi Manyama, May 09 2022

Keywords

Crossrefs

Cf. A052830, A353880, A353881.
Cf. A346923, A353885.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(x*log(1-x))^4/576)))
  • PARI
    a(n) = n!*sum(k=0, n\8, (4*k)!*abs(stirling(n-4*k, 4*k, 1))/(576^k*(n-4*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/8)} (4*k)! * |Stirling1(n-4*k,4*k)|/(576^k * (n-4*k)!).

A353883 Expansion of e.g.f. 1/(1 - (x * (exp(x) - 1))^2 / 4).

Original entry on oeis.org

1, 0, 0, 0, 6, 30, 105, 315, 3388, 47628, 497115, 4172025, 37829946, 491971194, 7699457857, 114432747975, 1602464966040, 23767387469688, 408590795439351, 7756561553900085, 149537297087139910, 2889288053301888630, 58297667473293537597
Offset: 0

Views

Author

Seiichi Manyama, May 09 2022

Keywords

Crossrefs

Cf. A052848, A353884, A353885.
Cf. A330047, A353880.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(x*(exp(x)-1))^2/4)))
  • PARI
    a(n) = n!*sum(k=0, n\4, (2*k)!*stirling(n-2*k, 2*k, 2)/(4^k*(n-2*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (2*k)! * Stirling2(n-2*k,2*k)/(4^k * (n-2*k)!).

A353891 Expansion of e.g.f. exp( (x * log(1-x))^2 / 4 ).

Original entry on oeis.org

1, 0, 0, 0, 6, 30, 165, 1050, 8932, 86184, 909360, 10393020, 129313206, 1743627600, 25314159780, 393346535400, 6512022804960, 114430467296880, 2127154061337480, 41703621476302800, 859966710771029040, 18606040434320713920, 421427283751799685360
Offset: 0

Views

Author

Seiichi Manyama, May 09 2022

Keywords

Crossrefs

Cf. A066166, A353892, A353893.
Cf. A347001, A353880, A353894.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x*log(1-x))^2/4)))
  • PARI
    a(n) = n!*sum(k=0, n\4, (2*k)!*abs(stirling(n-2*k, 2*k, 1))/(4^k*k!*(n-2*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (2*k)! * |Stirling1(n-2*k,2*k)|/(4^k * k! * (n-2*k)!).
Showing 1-5 of 5 results.

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