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 27 results. Next

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

A370988 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*(exp(x) - 1)) ).

Original entry on oeis.org

1, 0, 2, 3, 76, 425, 10326, 119077, 3158968, 57929265, 1740086290, 44066266541, 1512768107940, 48660920528233, 1905202422005806, 73878129769929045, 3275941116578461936, 147981592692778718561, 7366814796135956094378, 378666415166758834858237
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A052894, A370989, A370990.
Cf. A370991, A370992.
Cf. A052848.

Programs

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

Formula

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

A353999 Expansion of e.g.f. 1/(1 - x^3/6 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 20, 35, 1176, 10164, 58920, 277365, 3363580, 47567806, 519759604, 4591587455, 51017687280, 786120055400, 12187597925136, 165128862881769, 2261843835692340, 36940778814100210, 678763188831800380, 12143893591131411571, 211404290379223149384
Offset: 0

Views

Author

Seiichi Manyama, May 13 2022

Keywords

Crossrefs

Cf. A052848, A353998.
Cf. A000292, A351506, A354001.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^3/6*(exp(x)-1))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!/6*sum(j=4, i, 1/(j-3)!*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\4, k!*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));

Formula

a(0) = 1; a(n) = n!/6 * Sum_{k=4..n} 1/(k-3)! * a(n-k)/(n-k)! = binomial(n,3) * Sum_{k=4..n} binomial(n-3,k-3) * a(n-k).
a(n) = n! * Sum_{k=0..floor(n/4)} k! * Stirling2(n-3*k,k)/(6^k * (n-3*k)!).

A353998 Expansion of e.g.f. 1/(1 - x^2/2 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 3, 6, 10, 195, 1281, 5908, 68076, 758565, 6486535, 75598446, 1059484218, 13378016743, 185273328345, 2999003869800, 48665352612376, 816394913567433, 15110162148144267, 292156921946387170, 5805684093139498470, 122617308231635240331
Offset: 0

Views

Author

Seiichi Manyama, May 13 2022

Keywords

Crossrefs

Cf. A052848, A353999.
Cf. A351505, A354000.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^2/2*(exp(x)-1))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!/2*sum(j=3, i, 1/(j-2)!*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\3, k!*stirling(n-2*k, k, 2)/(2^k*(n-2*k)!));

Formula

a(0) = 1; a(n) = n!/2 * Sum_{k=3..n} 1/(k-2)! * a(n-k)/(n-k)! = binomial(n,2) * Sum_{k=3..n} binomial(n-2,k-2) * a(n-k).
a(n) = n! * Sum_{k=0..floor(n/3)} k! * Stirling2(n-2*k,k)/(2^k * (n-2*k)!).
a(n) ~ 2 * n! / ((4 + 2*r + r^3) * r^n), where r = 1.043121496712693605897520269472163423276582653660720448... is the root of the equation (exp(r)-1)*r^2 = 2. - Vaclav Kotesovec, May 13 2022

A358013 Expansion of e.g.f. 1/(1 - x^2 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 750, 5082, 23576, 453672, 5755770, 50894030, 841270452, 14694142476, 201442729670, 3552604015170, 73814245552560, 1369932831933392, 27860865121662066, 655240785723048726, 15052226249248287500, 357713461766745539700, 9416426612423343023742
Offset: 0

Views

Author

Seiichi Manyama, Oct 24 2022

Keywords

Crossrefs

Cf. A052848, A358014.
Cf. A240989, A351503, A353998.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^2*(exp(x)-1))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!*sum(j=3, i, 1/(j-2)!*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\3, k!*stirling(n-2*k, k, 2)/(n-2*k)!);

Formula

a(0) = 1; a(n) = n! * Sum_{k=3..n} 1/(k-2)! * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/3)} k! * Stirling2(n-2*k,k)/(n-2*k)!.

A358014 Expansion of e.g.f. 1/(1 - x^3 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 120, 210, 40656, 363384, 2117520, 9980190, 520250280, 9496208436, 109522054824, 982593614730, 28426015541280, 762523155318000, 14192088961120416, 204618562767970614, 4906638448867994040, 154037798077765359660, 4000484484370905087480
Offset: 0

Views

Author

Seiichi Manyama, Oct 24 2022

Keywords

Crossrefs

Cf. A052848, A358013.
Cf. A292891, A351504, A353999.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1-x^3 (Exp[x]-1)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 26 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^3*(exp(x)-1))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!*sum(j=4, i, 1/(j-3)!*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\4, k!*stirling(n-3*k, k, 2)/(n-3*k)!);

Formula

a(0) = 1; a(n) = n! * Sum_{k=4..n} 1/(k-3)! * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/4)} k! * Stirling2(n-3*k,k)/(n-3*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)!).

A353884 Expansion of e.g.f. 1/(1 - (x * (exp(x) - 1))^3 / 36).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 20, 210, 1400, 7560, 36120, 159390, 1035100, 17082780, 329893564, 5336661330, 73265956400, 889068944400, 9968073461616, 112902000191334, 1531070090032500, 27610559023112100, 586336131631313140, 12550716321612658266, 254052845940651258600
Offset: 0

Views

Author

Seiichi Manyama, May 09 2022

Keywords

Crossrefs

Cf. A052848, A353883, A353885.
Cf. A346894, A353881.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(x*(exp(x)-1))^3/36)))
  • PARI
    a(n) = n!*sum(k=0, n\6, (3*k)!*stirling(n-3*k, 3*k, 2)/(36^k*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/6)} (3*k)! * Stirling2(n-3*k,3*k)/(36^k * (n-3*k)!).

A353885 Expansion of e.g.f. 1/(1 - (x * (exp(x) - 1))^4 / 576).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 70, 1260, 13650, 115500, 841995, 5555550, 34139105, 198948750, 1175994820, 10315705400, 192609389700, 4563951046200, 98992258506345, 1898260633492650, 32787422848455275, 520556451785466250, 7722233521138092726, 108688302800107222500
Offset: 0

Views

Author

Seiichi Manyama, May 09 2022

Keywords

Crossrefs

Cf. A052848, A353883, A353884.
Cf. A346895, A353882.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(x*(exp(x)-1))^4/576)))
  • PARI
    a(n) = n!*sum(k=0, n\8, (4*k)!*stirling(n-4*k, 4*k, 2)/(576^k*(n-4*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/8)} (4*k)! * Stirling2(n-4*k,4*k)/(576^k * (n-4*k)!).

A367881 Expansion of e.g.f. 1/(1 - 3 * x * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 6, 9, 228, 1095, 23238, 215481, 4657992, 66216555, 1553967210, 29793656013, 777115661292, 18608934688383, 542832959656302, 15470567460571905, 503794462155308688, 16557037363336856019, 598704921471691072242, 22205328374455141122165
Offset: 0

Views

Author

Seiichi Manyama, Dec 03 2023

Keywords

Crossrefs

Cf. A052848, A367880.

Programs

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

Formula

a(0) = 1; a(n) = 3 * n * Sum_{k=2..n} binomial(n-1,k-1) * a(n-k).
a(n) = n! * Sum_{k=0..floor(n/2)} 3^k * k! * Stirling2(n-k,k)/(n-k)!.
Showing 1-10 of 27 results. Next

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