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.

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

A368173 Expansion of e.g.f. -log(1 - x^2/2 * (exp(x) - 1)).

Original entry on oeis.org

0, 0, 0, 3, 6, 10, 105, 651, 2968, 26496, 265905, 2203795, 22830456, 288661308, 3476579197, 44960585775, 671394654960, 10329701480416, 164573071219233, 2865785889662019, 52647629639499280, 1000194250108913580, 20125846165307543661, 426789766980101676943
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2023

Keywords

Links

Crossrefs

Cf. A052858, A368174.
Cf. A353998.

Programs

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

Formula

a(n) = n! * Sum_{k=1..floor(n/3)} (k-1)! * Stirling2(n-2*k,k)/(2^k * (n-2*k)!).
a(0) = a(1) = a(2) = 0; a(n) = n*(n-1)/2 + Sum_{k=3..n-1} k*(k-1)/2 * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Jan 22 2025

A368174 Expansion of e.g.f. -log(1 - x^3/6 * (exp(x) - 1)).

Original entry on oeis.org

0, 0, 0, 0, 4, 10, 20, 35, 616, 5124, 29520, 138765, 1312300, 16576846, 175795984, 1539037955, 15687832720, 216382727240, 3170822906976, 42007311638169, 553841577209940, 8435274815148370, 145708900713412960, 2517047758252082671, 42575155321545439384
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2023

Keywords

Comments

This sequence is different from A354001.

Links

Crossrefs

Cf. A052858, A368173.
Cf. A353999, A354001.

Programs

  • PARI
    a(n) = n!*sum(k=1, n\4, (k-1)!*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=1..floor(n/4)} (k-1)! * Stirling2(n-3*k,k)/(6^k * (n-3*k)!).
a(0) = a(1) = a(2) = a(3) = 0; a(n) = binomial(n,3) + Sum_{k=4..n-1} binomial(k,3) * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Jan 22 2025

A366564 Expansion of e.g.f. -log(1 - x^2 * (exp(x) - 1)).

Original entry on oeis.org

0, 0, 0, 6, 12, 20, 390, 2562, 11816, 166392, 1970730, 17131070, 241009692, 3861669396, 51411143966, 828234487290, 15865154629200, 283329069136112, 5431892804244306, 119420738547382134, 2628980439169097540, 59707303735169923980, 1488953374718002643142
Offset: 0

Views

Author

Seiichi Manyama, Dec 15 2023

Keywords

Crossrefs

Cf. A052858, A366751.
Cf. A366459, A368173.

Programs

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

Formula

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

A366751 Expansion of e.g.f. -log(1 - x^3 * (exp(x) - 1)).

Original entry on oeis.org

0, 0, 0, 0, 24, 60, 120, 210, 20496, 181944, 1059120, 4990590, 180292200, 3191349876, 36598884504, 327837512730, 7732754793120, 194896185648240, 3574721299186656, 51311061420097014, 1105883184455171640, 32127696556638165420, 812811279492629700360
Offset: 0

Views

Author

Seiichi Manyama, Dec 15 2023

Keywords

Crossrefs

Cf. A052858, A366564.
Cf. A368174.

Programs

  • PARI
    a(n) = n!*sum(k=1, n\4, (k-1)!*stirling(n-3*k, k, 2)/(n-3*k)!);

Formula

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

A380338 Expansion of e.g.f. log(1 - x * log(1 - x)).

Original entry on oeis.org

0, 0, 2, 3, -4, -30, 54, 1260, 3856, -36288, -279000, 2970000, 56725008, 109343520, -5495740992, -26086263840, 1293641890560, 21771049466880, -45508965806592, -4589738336217600, 10493846174810880, 2423866077943511040, 34328754265480012800, -358930542362135546880
Offset: 0

Views

Author

Seiichi Manyama, Jan 21 2025

Keywords

Crossrefs

Cf. A052804, A052858.
Cf. A089064, A380339.
Cf. A001048, A375684.

Programs

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

Formula

a(n) = n! * Sum_{k=1..floor(n/2)} (-1)^(k-1) * (k-1)! * |Stirling1(n-k,k)|/(n-k)!.
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).
Showing 1-6 of 6 results.

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