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 1-4 of 4 results.

A060487 Triangle T(n,k) of k-block tricoverings of an n-set (n >= 3, k >= 4).

Original entry on oeis.org

1, 3, 1, 7, 57, 95, 43, 3, 35, 717, 3107, 4520, 2465, 445, 12, 155, 7845, 75835, 244035, 325890, 195215, 50825, 4710, 70, 651, 81333, 1653771, 10418070, 27074575, 33453959, 20891962, 6580070, 965965, 52430, 465
Offset: 3

Views

Author

Vladeta Jovovic, Mar 20 2001

Keywords

Comments

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.

Examples

			Triangle begins:
  [1, 3, 1];
  [7, 57, 95, 43, 3];
  [35, 717, 3107, 4520, 2465, 445, 12];
  [155, 7845, 75835, 244035, 325890, 195215, 50825, 4710, 70];
  [651, 81333, 1653771, 10418070, 27074575, 33453959, 20891962, 6580070, 965965, 52430, 465];
   ...
There are 205 tricoverings of a 4-set(cf. A060486): 7 4-block, 57 5-block, 95 6-block, 43 7-block and 3 8-block tricoverings.

Links

Crossrefs

Columns include A060483, A060484, A060485.
Row sums are A060486.
Cf. A006095, A060090-A060095, A060069, A060070, A060051-A060053, A002718, A059443, A003462, A059945-A059951.

Programs

  • PARI
    WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); WeighT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}
row(n, k)={my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(y+x))); if(n==0, 1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])*y^(m-n)/(1+y))}
for(n=3, 8, print(Vecrev(row(3,n)))); \\ Andrew Howroyd, Dec 23 2018

Formula

E.g.f. for k-block tricoverings of an n-set is exp(-x+x^2/2+(exp(y)-1)*x^3/3)*Sum_{k=0..inf}x^k/k!*exp(-1/2*x^2*exp(k*y))*exp(binomial(k, 3)*y).

A060486 Tricoverings of an n-set.

Original entry on oeis.org

1, 0, 0, 5, 205, 11301, 904580, 101173251, 15207243828, 2975725761202, 738628553556470, 227636079973503479, 85554823285296622543, 38621481302086460057613, 20669385794052533823555309, 12966707189875262685801947906, 9441485712482676603570079314728
Offset: 0

Views

Author

Vladeta Jovovic, Mar 20 2001

Keywords

Comments

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.

Examples

			There are 1 4-block tricovering, 3 5-block tricoverings and 1 6-block tricovering of a 3-set (cf. A060487), so a(3)=5.

Links

Crossrefs

Row 3 of A188445.
Cf. A006095, A060483-A060485, (row sums of) A060487, A060090-A060095, A060069, A060070, A060051-A060053, A002718, A059443, A003462, A059945-A059951.

Formula

E.g.f. for k-block tricoverings of an n-set is exp(-x+x^2/2+(exp(y)-1)*x^3/3)*Sum_{k=0..inf}x^k/k!*exp(-1/2*x^2*exp(k*y))*exp(binomial(k, 3)*y).

Extensions

Terms a(11) and beyond from Andrew Howroyd, Dec 15 2018

A060483 Number of 5-block tricoverings of an n-set.

Original entry on oeis.org

3, 57, 717, 7845, 81333, 825237, 8300757, 83202645, 832809813, 8331237717, 83324947797, 833299785045, 8333199127893, 83332796486997, 833331185898837, 8333324743497045, 83333298973791573, 833333195894773077, 8333332783578305877, 83333331134311650645
Offset: 3

Views

Author

Vladeta Jovovic, Mar 20 2001

Keywords

Comments

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.

Links

Crossrefs

Column k=5 of A060487.
Cf. A006095, A060484, A060485, A060486, A060090-A060095, A060069, A060070, A060051-A060053, A002718, A059443, A003462, A059945-A059951.

Programs

Formula

a(n) = (1/5!)*(10^n - 15*4^n + 45*2^n - 40).
Generally, e.g.f. for k-block tricoverings of an n-set is exp(-x+x^2/2+(exp(y)-1)*x^3/3)*Sum_{k=0..inf}x^k/k!*exp(-1/2*x^2*exp(k*y))*exp(binomial(k, 3)*y).
G.f.: 3*x^3*(2*x+1) / ((x-1)*(2*x-1)*(4*x-1)*(10*x-1)). - Colin Barker, Jan 11 2013

Extensions

More terms from Colin Barker, Jan 11 2013

A060484 Number of 6-block tricoverings of an n-set.

Original entry on oeis.org

1, 95, 3107, 75835, 1653771, 34384875, 700030507, 14116715435, 283432939691, 5679127043755, 113683003777707, 2274630646577835, 45502044971338411, 910133025632152235, 18203564201836161707, 364080180268471397035
Offset: 3

Views

Author

Vladeta Jovovic, Mar 20 2001

Keywords

Comments

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.

Links

Crossrefs

Column k=6 of A060487.
Cf. A006095, A060483, A060485, A060486, A060090-A060095, A060069, A060070, A060051-A060053, A002718, A059443, A003462, A059945-A059951.

Programs

  • Mathematica
    With[{c=1/6!},Table[c(20^n-6*10^n-15*8^n+135*4^n-310*2^n+240),{n,3,20}]] (* or *) LinearRecurrence[{45,-720,5220,-17664,25920,-12800},{1,95,3107,75835,1653771,34384875},20] (* Harvey P. Dale, Jan 05 2017 *)
  • PARI
    a(n) = (1/6!)*(20^n - 6*10^n - 15*8^n + 135*4^n - 310*2^n + 240) \\ Andrew Howroyd, Dec 15 2018

Formula

a(n) = (1/6!)*(20^n - 6*10^n - 15*8^n + 135*4^n - 310*2^n + 240).
E.g.f. for k-block tricoverings of an n-set is exp(-x+x^2/2+(exp(y)-1)*x^3/3)*Sum_{k=0..inf}x^k/k!*exp(-1/2*x^2*exp(k*y))*exp(binomial(k, 3)*y).
G.f.: -x^3*(800*x^3+448*x^2-50*x-1) / ((x-1)*(2*x-1)*(4*x-1)*(8*x-1)*(10*x-1)*(20*x-1)). - Colin Barker, Jan 12 2013
a(n) = 45*a(n-1)-720*a(n-2)+5220*a(n-3)-17664*a(n-4)+25920*a(n-5)-12800*a(n-6). - Wesley Ivan Hurt, Oct 18 2021
Showing 1-4 of 4 results.

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