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 30 results. Next

A370993 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + x*log(1-x)) ).

Original entry on oeis.org

1, 0, 2, 3, 80, 450, 11424, 133140, 3670400, 68303088, 2123212320, 54742984560, 1938915574848, 63653459126400, 2565847637273088, 101718189575664480, 4637150408792355840, 214393171673968519680, 10962579011721928980480, 577166004742408670937600
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A052802, A370994, A370995.
Cf. A052830, A370988.

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (n+k)! * |Stirling1(n-k,k)|/(n-k)!.

A351503 Expansion of e.g.f. 1/(1 + x^2 * log(1 - x)).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 900, 6048, 43680, 717120, 8658720, 102231360, 1735525440, 28819964160, 473955850368, 9235543363200, 189202617676800, 3940225003653120, 89804740509434880, 2169337606086389760, 54085753764912844800, 1429100881569205125120
Offset: 0

Views

Author

Seiichi Manyama, May 04 2022

Keywords

Links

Crossrefs

Cf. A052830, A351504.
Cf. A351505, A353228.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1+x^2 Log[1-x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 18 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+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!*sum(j=3, i, 1/(j-2)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\3, k!*abs(stirling(n-2*k, k, 1))/(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! * |Stirling1(n-2*k,k)|/(n-2*k)!.

A351505 Expansion of e.g.f. 1/(1 + x^2/2 * log(1 - x)).

Original entry on oeis.org

1, 0, 0, 3, 6, 20, 270, 1764, 12600, 146880, 1597680, 17934840, 243777600, 3506518080, 52696595952, 870564618000, 15354480960000, 284780747946240, 5622461683666560, 117425971162442880, 2574172644658272000, 59302473667128599040, 1432738540209781728000
Offset: 0

Views

Author

Seiichi Manyama, May 04 2022

Keywords

Links

Crossrefs

Cf. A052830, A351506.
Cf. A351492, A351503,

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x^2/2*log(1-x))))
  • 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!*abs(stirling(n-2*k, k, 1))/(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)!.
a(n) = n! * Sum_{k=0..floor(n/3)} k! * |Stirling1(n-2*k,k)|/(2^k * (n-2*k)!).

A351504 Expansion of e.g.f. 1/(1 + x^3 * log(1 - x)).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 240, 1260, 48384, 423360, 3844800, 38253600, 896797440, 14322147840, 216997522560, 3350656108800, 74820944056320, 1621271286835200, 34293811249152000, 727304513980262400, 18147791755697356800, 476653146551318016000
Offset: 0

Views

Author

Seiichi Manyama, May 04 2022

Keywords

Links

Crossrefs

Cf. A052830, A351503.
Cf. A351506, A353229.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+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!*sum(j=4, i, 1/(j-3)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\4, k!*abs(stirling(n-3*k, k, 1))/(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! * |Stirling1(n-3*k,k)|/(n-3*k)!.

A351506 Expansion of e.g.f. 1/(1 + x^3/6 * log(1 - x)).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 40, 210, 2464, 20160, 178800, 1755600, 21215040, 268107840, 3596916960, 51452200800, 800489733120, 13262804755200, 232536822336000, 4300843392518400, 84023034413644800, 1727339274045504000, 37248117171719731200, 840387048760633651200
Offset: 0

Views

Author

Seiichi Manyama, May 04 2022

Keywords

Links

Crossrefs

Cf. A052830, A351505.
Cf. A351493, A351504,

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x^3/6*log(1-x))))
  • 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!*abs(stirling(n-3*k, k, 1))/(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)!.
a(n) = n! * Sum_{k=0..floor(n/4)} k! * |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).

A052804 A simple grammar: cycles of rooted cycles.

Original entry on oeis.org

0, 0, 2, 3, 20, 90, 714, 5460, 54704, 580608, 7214040, 96932880, 1452396912, 23507621280, 414102201408, 7827185489760, 158757800613120, 3429996441661440, 78775916315263488, 1914627403408320000, 49126748261368331520, 1326584986873331189760
Offset: 0

Views

Author

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

Keywords

Links

Crossrefs

Cf. A052830, A052858.

Programs

  • Maple
    spec := [S,{B=Prod(C,Z),C=Cycle(Z),S=Cycle(B)},labeled]: seq(combstruct[count](spec, size=n), n=0..20);
  • Mathematica
    nn = 25; Range[0, nn]! CoefficientList[Series[Log[-1/(-1 + Log[-1/(-1 + x)]*x)], {x, 0, nn}], x] (* T. D. Noe, Feb 21 2013 *)
  • PARI
    N = 66;  x = 'x + O('x^N);
egf = -log(1 + x*log(1-x)) + 'c0;
gf = serlaplace(egf);
v = Vec(gf);  v[1]-='c0;  v
/* Joerg Arndt, Feb 21 2013 */

Formula

E.g.f.: log(-1/(-1+log(-1/(-1+x))*x)).
E.g.f.: -log(1 + x*log(1-x)). - Arkadiusz Wesolowski, Feb 21 2013
a(n) ~ (n-1)! * r^n, where r = 1.349976485401125... is the root of the equation (r-1)*exp(r) = r. - Vaclav Kotesovec, Oct 01 2013
a(n) = n! * Sum_{k=1..floor(n/2)} (k-1)! * |Stirling1(n-k,k)|/(n-k)!. - Seiichi Manyama, Dec 13 2023
a(0) = a(1) = 0; a(n) = n * (n-2)! + Sum_{k=2..n-1} k * (k-2)! * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Jan 21 2025

A353880 Expansion of e.g.f. 1/(1 - (x * log(1-x))^2 / 4).

Original entry on oeis.org

1, 0, 0, 0, 6, 30, 165, 1050, 10192, 108864, 1230660, 14758920, 195861996, 2852815680, 44880446520, 753211040400, 13458760362720, 255688784416800, 5149255813778160, 109489194918180000, 2450182706364430080, 57567025900160259840, 1417073899136197468320
Offset: 0

Views

Author

Seiichi Manyama, May 09 2022

Keywords

Crossrefs

Cf. A052830, A353881, A353882.
Cf. A346921, A353883.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(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*(n-2*k)!));

Formula

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

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)!).

A375167 Expansion of e.g.f. 1 / (1 + x * log(1 - x^2/2)).

Original entry on oeis.org

1, 0, 0, 3, 0, 15, 180, 210, 5040, 51030, 207900, 3991680, 42411600, 356756400, 6485398920, 80635054500, 1040690851200, 19440077857200, 291313362740400, 4914773560897200, 98182334033784000, 1763213788027692000, 35636304386103220800, 778379605589616030000
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2024

Keywords

Crossrefs

Cf. A052830, A375556, A375558.
Cf. A351155.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)! * |Stirling1(k,n-2*k)|/(2^k*k!).
Showing 1-10 of 30 results. Next

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

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