A038158 -id:A038158 - cp's OEIS frontend

A268217 Triangle read by rows: T(n,k) (n>=3, k=3..n) is the number of topologies t on n points having exactly k open sets such that t contains exactly one open set of size m for each m in {0,2,3,4,...,s,n} where s is the size of the largest proper open set in t.

3, 6, 12, 10, 30, 60, 15, 60, 180, 360, 21, 105, 420, 1260, 2520, 28, 168, 840, 3360, 10080, 20160, 36, 252, 1512, 7560, 30240, 90720, 181440, 45, 360, 2520, 15120, 75600, 302400, 907200, 1814400, 55, 495, 3960, 27720, 166320, 831600, 3326400, 9979200, 19958400

Offset: 3

N. J. A. Sloane, Jan 29 2016

When two leading 0's are added and last element repeated, rows give the coefficients of the path polynomials of the complete graph K_n. - Eric W. Weisstein, Jun 04 2017

			Triangle begins:
   3;
   6,  12;
  10,  30,   60;
  15,  60,  180,   360;
  21, 105,  420,  1260,   2520;
  28, 168,  840,  3360,  10080,  20160;
  36, 252, 1512,  7560,  30240,  90720,  181440;
  45, 360, 2520, 15120,  75600, 302400,  907200, 1814400;
  ...

Andrew Howroyd, Table of n, a(n) for n = 3..1277 (first 50 rows)
G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.
Eric Weisstein's World of Mathematics, Complete Graph.
Eric Weisstein's World of Mathematics, Graph Path.

Row sums give A038158.
Triangles in this series: A119741, A268217, A268221, A268222, A268223.
Cf. A282507.

i = 2; Table[Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0, n - i - 1}], {n, 2, 9}] // Grid (* Geoffrey Critzer, Feb 19 2017 *)
CoefficientList[Table[-(1/2) (n - 1) n x^(n - 2) (Gamma[n - 1] - E^(1/x) Gamma[n - 1, 1/x]), {n, 3, 10}] // FunctionExpand, x] // Flatten (* Eric W. Weisstein, Jun 04 2017 *)

Title clarified by Geoffrey Critzer, Feb 19 2017

Corrected and extended by Andrew Howroyd, Aug 09 2025

A268219 a(n) = (n!/4!)*Sum(1/k!,k=1..n-4).

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 45, 350, 2870, 25956, 259770, 2857800, 34294095, 445823950, 6241536301, 93623045880, 1497968735900, 25465468512680, 458378433231300, 8709190231398576, 174183804627976365, 3657859897187509650, 80472917738125219615, 1850877107976880060000, 44421050591445121450626

Offset: 0

N. J. A. Sloane, Jan 30 2016

nonn

G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.

For others in this series, see A038156, A038158, A268218, A268220.

g:=(r,n)->(n!/r!)*add(1/k!,k=1..n-r);
g2:=r->[seq(g(r,n),n=0..30)];
g2(4);

a(n) = (n!/4!)*sum(k=1, n-4, 1/k!); \\ Michel Marcus, Jan 30 2016

A268218 a(n) = (n!/3!)*Sum(1/k!,k=1..n-3).

Original entry on oeis.org

0, 0, 0, 0, 4, 30, 200, 1435, 11536, 103908, 1039200, 11431365, 137176600, 1783296086, 24966145568, 374492183975, 5991874944160, 101861874051400, 1833513732926016, 34836760925595273, 696735218511906600, 14631439588750039930, 321891670952500880000, 7403508431907520241771

Offset: 0

N. J. A. Sloane, Jan 30 2016

nonn

G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.

For others in this series, see A038156, A038158, A268219, A268220.

g:=(r,n)->(n!/r!)*add(1/k!,k=1..n-r);
g2:=r->[seq(g(r,n),n=0..30)];
g2(3);

a(n) = (n!/3!)*sum(k=1, n-3, 1/k!); \\ Michel Marcus, Jan 30 2016

A268220 a(n) = (n!/5!)*Sum(1/k!,k=1..n-5).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 6, 63, 560, 5166, 51912, 571494, 6858720, 89164647, 1248307060, 18724608903, 299593746816, 5093093702060, 91675686645648, 1741838046278940, 34836760925594304, 731571979437500733, 16094583547625042460, 370175421595376010229, 8884210118289024288000

Offset: 0

N. J. A. Sloane, Jan 30 2016

nonn

G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.

For others in this series, see A038156, A038158, A268218, A268219.

g:=(r,n)->(n!/r!)*add(1/k!,k=1..n-r);
g2:=r->[seq(g(r,n),n=0..30)];
g2(5);

a(n) = (n!/5!)*sum(k=1, n-5, 1/k!); \\ Michel Marcus, Jan 30 2016

cp's OEIS Frontend

A268217 Triangle read by rows: T(n,k) (n>=3, k=3..n) is the number of topologies t on n points having exactly k open sets such that t contains exactly one open set of size m for each m in {0,2,3,4,...,s,n} where s is the size of the largest proper open set in t.

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A268219 a(n) = (n!/4!)*Sum(1/k!,k=1..n-4).

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A268218 a(n) = (n!/3!)*Sum(1/k!,k=1..n-3).

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Maple

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A268220 a(n) = (n!/5!)*Sum(1/k!,k=1..n-5).

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Maple

PARI