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 24 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

A007113 Expansion of e.g.f. (1 + x)^x.

Original entry on oeis.org

1, 0, 2, -3, 20, -90, 594, -4200, 34544, -316008, 3207240, -35699400, 432690312, -5672581200, 79991160144, -1207367605080, 19423062612480, -331770360922560, 5997105160795584, -114373526841360000, 2295170834453089920, -48344592370577247360
Offset: 0

Views

Author

Simon Plouffe

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.3.

Links

Crossrefs

Cf. A053489, A053490. Apart from initial terms and signs, same as A066166.

Programs

  • Maple
    a:= n-> n! *coeff(series((1+x)^x, x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 12 2012
  • Mathematica
    CoefficientList[Series[(1 + x)^x, {x, 0, 19}], x]*Table[(n - 1)!, {n, 1, 20}]
a[n_] := (-1)^n*n!*Sum[ StirlingS1[n - k, k]/(n - k)!*(-1)^(n - 2*k), {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Dec 12 2012, after Vladeta Jovovic *)

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*k!*Stirling1(n-k, k). - Vladeta Jovovic, Dec 19 2004
a(n) ~ (-1)^n * n!. - Vaclav Kotesovec, Jun 06 2019

Extensions

Signs from Christian G. Bower, Nov 15 1998

A353228 Expansion of e.g.f. (1 - x)^(-x^2).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 540, 3528, 25200, 263520, 2741760, 30048480, 372794400, 4971957120, 70612686144, 1076056027200, 17469796780800, 300562292459520, 5468568356666880, 104917700221125120, 2116572758902425600, 44794683422986936320, 992435268252158438400
Offset: 0

Views

Author

Seiichi Manyama, May 01 2022

Keywords

Links

Crossrefs

Cf. A066166, A351503, A351492, A353223, A353229.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1-x)^(-x^2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, May 12 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-x^2)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x^2*log(1-x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, j/(j-2)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\3, abs(stirling(n-2*k, k, 1))/(n-2*k)!);

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k/(k-2) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/3)} |Stirling1(n-2*k,k)|/(n-2*k)!.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / exp(n). - Vaclav Kotesovec, May 04 2022

A353229 Expansion of e.g.f. (1 - x)^(-x^3).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 240, 1260, 28224, 241920, 2181600, 21621600, 315342720, 4358914560, 61607407680, 912518006400, 15142006978560, 265601118182400, 4877947688140800, 93691850626483200, 1901787789077452800, 40548028309147699200, 904101131200045363200
Offset: 0

Views

Author

Seiichi Manyama, May 01 2022

Keywords

Links

Crossrefs

Cf. A066166, A351493, A351504, A353228.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[(1-x)^-x^3,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 20 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-x^3)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x^3*log(1-x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=4, i, j/(j-3)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\4, abs(stirling(n-3*k, k, 1))/(n-3*k)!);

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=4..n} k/(k-3) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/4)} |Stirling1(n-3*k,k)|/(n-3*k)!.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / exp(n). - Vaclav Kotesovec, May 04 2022

A355610 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - x)^(-x^k/k!).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 2, 6, 1, 0, 0, 3, 24, 1, 0, 0, 3, 20, 120, 1, 0, 0, 0, 6, 90, 720, 1, 0, 0, 0, 4, 20, 594, 5040, 1, 0, 0, 0, 0, 10, 180, 4200, 40320, 1, 0, 0, 0, 0, 5, 40, 1134, 34544, 362880, 1, 0, 0, 0, 0, 0, 15, 210, 7980, 316008, 3628800, 1, 0, 0, 0, 0, 0, 6, 70, 1904, 71280, 3207240, 39916800
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2022

Keywords

Examples

			Square array begins:
    1,   1,   1,  1,  1, 1, 1, ...
    1,   0,   0,  0,  0, 0, 0, ...
    2,   2,   0,  0,  0, 0, 0, ...
    6,   3,   3,  0,  0, 0, 0, ...
   24,  20,   6,  4,  0, 0, 0, ...
  120,  90,  20, 10,  5, 0, 0, ...
  720, 594, 180, 40, 15, 6, 0, ...

Links

Crossrefs

Columns k=0..4 give A000142, A066166, A351492, A351493, A355507.
Cf. A355609, A355619.

Programs

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

Formula

T(0,k) = 1 and T(n,k) = (n-1)!/k! * Sum_{j=k+1..n} j/(j-k) * T(n-j,k)/(n-j)! for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} |Stirling1(n-k*j,j)|/(k!^j * (n-k*j)!).

A053491 Expansion of e.g.f. (1-2*x)^(-x).

Original entry on oeis.org

1, 0, 4, 12, 112, 960, 10848, 141120, 2122496, 36094464, 685578240, 14385761280, 330532435968, 8253827112960, 222587077558272, 6447285982126080, 199630453605335040, 6580280144225894400, 230056747973625249792, 8503148524089755566080
Offset: 0

Views

Author

N. J. A. Sloane, Jan 15 2000

Keywords

Links

Crossrefs

Cf. A007113, A053489, A066166.

Programs

  • Mathematica
    CoefficientList[Series[(1-2x)^(-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)
  • PARI
    a(n) = n!*sum(k=0, n\2, 2^(n-k)*abs(stirling(n-k, k, 1))/(n-k)!); \\ Seiichi Manyama, May 20 2022

Formula

a(n) ~ 2^(n+1/2)*n^n/exp(n). - Vaclav Kotesovec, Jun 27 2013
a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-k) * |Stirling1(n-k,k)|/(n-k)!. - Seiichi Manyama, May 20 2022

A353226 Expansion of e.g.f. (1 - x^2)^(-x).

Original entry on oeis.org

1, 0, 0, 6, 0, 60, 360, 1680, 20160, 151200, 1663200, 17962560, 219542400, 2854051200, 40441040640, 606356150400, 9793028044800, 166481476761600, 3017626733721600, 57359043873331200, 1153275200453376000, 24233844054131712000, 535361100608439705600
Offset: 0

Views

Author

Seiichi Manyama, May 01 2022

Keywords

Crossrefs

Cf. A066166, A121452, A351155, A353227.

Programs

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

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=2..floor((n+1)/2)} (2*k-1)/(k-1) * a(n-2*k+1)/(n-2*k+1)!.
a(n) = n! * Sum_{k=0..floor(n/2)} |Stirling1(k,n-2*k)|/k!.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (2*exp(n)). - Vaclav Kotesovec, May 04 2022

A355064 Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k)^(1/k) )^x.

Original entry on oeis.org

1, 0, 2, 6, 28, 210, 1248, 13020, 102128, 1248912, 13457880, 176726880, 2362784928, 36609693120, 551337892896, 9588702417840, 171779733546240, 3230529997766400, 64714946343904512, 1371420774325866240, 29953522454811096960, 698447624328756610560
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2022

Keywords

Links

Crossrefs

Cf. A354623, A356554.
Cf. A000005, A066166, A356336, A356564.

Programs

  • Mathematica
    a[0] := a[0] = 1; a[1] := a[1] = 0;
a[n_] := a[n] = Sum[Factorial[k]*DivisorSigma[0, k - 1]/(k - 1)*Binomial[n - 1, k - 1]* a[n - k], {k, 2, n}];
Table[a[n], {n, 0, 50}] (* Sidney Cadot, Jan 05 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-x^k)^(1/k))^x))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*sigma(j-1, 0)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=2..n} k! * sigma_0(k-1)/(k-1) * binomial(n-1,k-1) * a(n-k).

A355609 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - x)^(-x^k).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 2, 6, 1, 0, 0, 3, 24, 1, 0, 0, 6, 20, 120, 1, 0, 0, 0, 12, 90, 720, 1, 0, 0, 0, 24, 40, 594, 5040, 1, 0, 0, 0, 0, 60, 540, 4200, 40320, 1, 0, 0, 0, 0, 120, 240, 3528, 34544, 362880, 1, 0, 0, 0, 0, 0, 360, 1260, 25200, 316008, 3628800, 1, 0, 0, 0, 0, 0, 720, 1680, 28224, 263520, 3207240, 39916800
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2022

Keywords

Examples

			Square array begins:
    1,   1,   1,   1,   1,   1, 1, ...
    1,   0,   0,   0,   0,   0, 0, ...
    2,   2,   0,   0,   0,   0, 0, ...
    6,   3,   6,   0,   0,   0, 0, ...
   24,  20,  12,  24,   0,   0, 0, ...
  120,  90,  40,  60, 120,   0, 0, ...
  720, 594, 540, 240, 360, 720, 0, ...

Links

Crossrefs

Columns k=0..4 give A000142, A066166, A353228, A353229, A354624.
Cf. A355607, A355610.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n\(k+1), abs(stirling(n-k*j, j, 1))/(n-k*j)!);

Formula

T(0,k) = 1 and T(n,k) = (n-1)! * Sum_{j=k+1..n} j/(j-k) * T(n-j,k)/(n-j)! for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} |Stirling1(n-k*j,j)|/(n-k*j)!.

A353227 Expansion of e.g.f. (1 - x^3)^(-x).

Original entry on oeis.org

1, 0, 0, 0, 24, 0, 0, 2520, 20160, 0, 1209600, 19958400, 79833600, 1556755200, 39956716800, 326918592000, 5056340889600, 148203095040000, 1867358997504000, 30411275102208000, 946128558735360000, 15965919428659200000, 293266062902292480000
Offset: 0

Views

Author

Seiichi Manyama, May 01 2022

Keywords

Crossrefs

Cf. A066166, A351156, A353223, A353226.

Programs

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

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=2..floor((n+2)/3)} (3*k-2)/(k-1) * a(n-3*k+2)/(n-3*k+2)!.
a(n) = n! * Sum_{k=0..floor(n/3)} |Stirling1(k,n-3*k)|/k!.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (3*exp(n)). - Vaclav Kotesovec, May 04 2022
Showing 1-10 of 24 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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