A059443 -id:A059443 - cp's OEIS frontend

A060486 Tricoverings of an n-set.

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

Offset: 0

Vladeta Jovovic, Mar 20 2001

nonn

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

			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.

Andrew Howroyd, Table of n, a(n) for n = 0..100

Row 3 of A188445.

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

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).

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

A060492 Triangle T(n,k) of k-block ordered tricoverings of an unlabeled n-set (n >= 3, k = 4..2n).

Original entry on oeis.org

4, 60, 120, 13, 375, 3030, 9030, 5040, 28, 1392, 24552, 207900, 838320, 1345680, 362880, 50, 4020, 130740, 2208430, 20334720, 101752560, 257065200, 261122400, 46569600, 80, 9960, 551640, 16365410, 274814760, 2709457128, 15812198640

Offset: 3

Vladeta Jovovic, Mar 20 2001

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

All columns are polynomials of order binomial(k, 3). - Andrew Howroyd, Jan 30 2020

			Triangle begins:
  [4, 60, 120],
  [13, 375, 3030, 9030, 5040],
  [28, 1392, 24552, 207900, 838320, 1345680, 362880],
  [50, 4020, 130740, 2208430, 20334720, 101752560, 257065200, 261122400, 46569600], [80, 9960, 551640, 16365410, 274814760, 2709457128, 15812198640, 52897521600, 91945022400, 64778313600, 8043235200],
   ...
There are 184 ordered tricoverings of an unlabeled 3-set: 4 4-block, 60 5-block and 120 6-block tricoverings (cf. A060491).

Andrew Howroyd, Table of n, a(n) for n = 3..1522 (rows n=3..40)

Row sums are A060491.
Columns k=4..6 are A060488, A060489, A060490.
Cf. A059443, A059530, A060052, A060092, A060487, A331571.

\\ gives g.f. of k-th column.
ColGf(k) = k!*polcoef(exp(-x + x^2/2 + x^3*y/(3*(1-y)) + O(x*x^k) )*sum(j=0, k, 1/(1-y)^binomial(j, 3)*exp((-x^2/2)/(1-y)^j + O(x*x^k))*x^j/j!), k) \\ Andrew Howroyd, Jan 30 2020

T(n)={my(m=2*n, y='y + O('y^(n+1))); my(g=serlaplace(exp(-x + x^2/2 + x^3*y/(3*(1-y)) + O(x*x^m))*sum(k=0, m, 1/(1-y)^binomial(k, 3)*exp((-x^2/2)/(1-y)^k + O(x*x^m))*x^k/k!))); Mat([Col(p/y^3, -n) | p<-Vec(g)[2..m+1]])}
{ my(A=T(8)); for(n=3, matsize(A)[1], print(A[n, 4..2*n])) } \\ Andrew Howroyd, Jan 30 2020

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

A059530 Triangle T(n,k) of k-block T_0-tricoverings of an n-set, n >= 3, k = 0..2*n.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 1, 39, 89, 43, 3, 0, 0, 0, 0, 0, 252, 2192, 4090, 2435, 445, 12, 0, 0, 0, 0, 0, 1260, 37080, 179890, 289170, 188540, 50645, 4710, 70, 0, 0, 0, 0, 0, 5040, 536760, 6052730, 20660055, 29432319, 19826737, 6481160, 964495, 52430

Offset: 3

Vladeta Jovovic, Feb 22 2001

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering. A covering of a set is a T_0-covering if for every two distinct elements of the set there exists a block of the covering containing one but not the other element.

			Triangle begins:
  [0, 0, 0, 0, 1, 3, 1],
  [0, 0, 0, 0, 1, 39, 89, 43, 3],
  [0, 0, 0, 0, 0, 252, 2192, 4090, 2435, 445, 12],
  [0, 0, 0, 0, 0, 1260, 37080, 179890, 289170, 188540, 50645, 4710, 70],
   ...
There are 5 = 1 + 3 + 1 T_0-tricoverings of a 3-set and 175 = 1 + 39 + 89 + 43 + 3 T_0-tricoverings of a 4-set, cf. A060070.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

Andrew Howroyd, Table of n, a(n) for n = 3..1674
Vladeta Jovovic, T_0-tricoverings of a 4-set

Column sums are A060069.
Row sums are A060070.
Cf. A059443, A060052, A060092, A060487, A060492, A331039.

\\ gets k-th column as vector
C(k)=if(k<4, [], Vecrev(serlaplace(polcoef(exp(-x + x^2/2 + x^3*y/3 + O(x*x^k))*sum(i=0, 2*k, (1+y)^binomial(i, 3)*exp(-x^2*(1+y)^i/2 + O(x*x^k))*x^i/i!), k))/y)) \\ Andrew Howroyd, Jan 30 2020

T(n)={my(m=2*n, y='y + O('y^(n+1))); my(g=exp(-x + x^2/2 + x^3*y/3 + O(x*x^m))*sum(k=0, m, (1+y)^binomial(k, 3)*exp(-x^2*(1+y)^k/2 + O(x*x^m))*x^k/k!)); Mat([Col(serlaplace(p), -n) | p<-Vec(g)[2..m+1]]);}
{ my(A=T(8)); for(n=3, matsize(A)[1], print(concat([0], A[n, 1..2*n]))) } \\ Andrew Howroyd, Jan 30 2020

E.g.f. for k-block T_0-tricoverings of an n-set is exp(-x+1/2*x^2+1/3*x^3*y)*Sum_{i=0..inf}(1+y)^binomial(i, 3)*exp(-1/2*x^2*(1+y)^i)*x^i/i!.

T(n,k) = 0 for n > binomial(k, 3). - Andrew Howroyd, Jan 30 2020

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

Vladeta Jovovic, Mar 20 2001

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

Andrew Howroyd, Table of n, a(n) for n = 3..500
Index entries for linear recurrences with constant coefficients, signature (17,-84,148,-80).

Column k=5 of A060487.

Cf. A006095, A060484, A060485, A060486, A060090-A060095, A060069, A060070, A060051-A060053, A002718, A059443, A003462, A059945-A059951.

Vec(3*(1+2*x)/(x-1)/(2*x-1)/(4*x-1)/(10*x-1)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2013

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

More terms from Colin Barker, Jan 11 2013

A060491 Number of ordered tricoverings of an unlabeled n-set.

Original entry on oeis.org

1, 0, 0, 184, 17488, 2780752, 689187720, 236477490418, 107317805999204, 62318195302890305, 45081693413563797127, 39762626850034005271588, 42009504510315968282400843, 52381340312720286113688037624, 76118747309505733406576769607755

Offset: 0

Vladeta Jovovic, Mar 20 2001

nonn

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

			There are 184 ordered tricoverings of an unlabeled 3-set: 4 4-block, 60 5-block and 120 6-block tricoverings (cf. A060492).

Andrew Howroyd, Table of n, a(n) for n = 0..100

Row n=3 of A331571.
Row sums of A060492.
Cf. A060486, A060487, A060090, A060092, A060069, A060070, A060051, A060052, A060053, A002718, A059443.

seq(n)={my(m=2*n\2, y='y + O('y^(n+1))); Vec(subst(Pol(serlaplace(exp(-x + x^2/2 + x^3*y/(3*(1-y)) + O(x*x^m))*sum(k=0, m, 1/(1-y)^binomial(k, 3)*exp((-x^2/2)/(1-y)^k + O(x*x^m))*x^k/k!))), x, 1))} \\ Andrew Howroyd, Jan 30 2020

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

Terms a(11) and beyond from Andrew Howroyd, Jan 30 2020

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

Vladeta Jovovic, Mar 20 2001

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

Andrew Howroyd, Table of n, a(n) for n = 3..200
Index entries for linear recurrences with constant coefficients, signature (45,-720,5220,-17664,25920,-12800).

Column k=6 of A060487.

Cf. A006095, A060483, A060485, A060486, A060090-A060095, A060069, A060070, A060051-A060053, A002718, A059443, A003462, A059945-A059951.

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 *)

a(n) = (1/6!)*(20^n - 6*10^n - 15*8^n + 135*4^n - 310*2^n + 240) \\ Andrew Howroyd, Dec 15 2018

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

A060485 Number of 7-block tricoverings of an n-set.

Original entry on oeis.org

43, 4520, 244035, 10418070, 401861943, 14778678180, 530817413155, 18837147108890, 664260814445943, 23345018969140440, 818942064306004275, 28699514624047140510, 1005201938765467579543, 35196266296400319440300

Offset: 4

Vladeta Jovovic, Mar 20 2001

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

Andrew Howroyd, Table of n, a(n) for n = 4..200
Index entries for linear recurrences with constant coefficients, signature (110, -4991, 124120, -1887459, 18470550, -118758569, 501056740, -1355000500, 2223560000, -1973160000, 705600000).

Column k=7 of A060487.

Cf. A006095, A060483, A060484, A060486, A060090-A060095, A060069, A060070, A060051-A060053, A002718, A059443, A003462, A059945-A059951.

a(n) = (1/7!)*(35^n - 7*20^n - 21*15^n + 42*10^n + 105*8^n + 105*7^n + 70*5^n - 945*4^n - 525*3^n + 2450*2^n - 1470).
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..infinity}x^k/k!*exp(-1/2*x^2*exp(k*y))*exp(binomial(k, 3)*y).
G.f.: x^4*(27300000*x^7 +9288000*x^6 -17908650*x^5 +6008735*x^4 -796380*x^3 +38552*x^2 +210*x -43) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(7*x -1)*(8*x -1)*(10*x -1)*(15*x -1)*(20*x -1)*(35*x -1)). - Colin Barker, Jan 12 2013

A059946 Number of 5-block bicoverings of an n-set.

Original entry on oeis.org

0, 0, 0, 25, 472, 6185, 70700, 759045, 7894992, 80736625, 817897300, 8241325565, 82783813112, 830046591465, 8313655213500, 83215436364085, 832626645756832, 8329096006484705, 83307920631515300, 833180902353754605, 8332418928963358152, 83327847634888960345

Offset: 1

Vladeta Jovovic, Feb 14 2001

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (26,-255,1210,-2924,3384,-1440).

Column k=5 of A059443.

Cf. A002718.

With[{c=(1/5!)},Table[c(10^n-5 6^n-10 4^n+20 3^n+30 2^n-60),{n,20}]]  (* Harvey P. Dale, Apr 21 2011 *)

a(n) = {(1/5!)*(10^n - 5*6^n - 10*4^n + 20*3^n + 30*2^n - 60)} \\ Andrew Howroyd, Jan 29 2020

a(n) = (1/5!)*(10^n - 5*6^n - 10*4^n + 20*3^n + 30*2^n - 60).
E.g.f. for m-block bicoverings of an n-set is exp(-x-1/2*x^2*(exp(y)-1))*Sum_{i>=0} (x^i/i!)*exp(binomial(i, 2)*y).
G.f.: x^4*(288*x^2-178*x+25) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(6*x-1)*(10*x-1)). - Colin Barker, Jan 11 2013

More terms from Colin Barker, Jan 11 2013

A059947 Number of 6-block bicoverings of an n-set.

Original entry on oeis.org

0, 0, 0, 3, 256, 7255, 149660, 2681063, 44659776, 714287535, 11154475420, 171673613023, 2618246526896, 39701554817015, 599773397512380, 9038881598035383, 136004367641775616, 2044264589908169695, 30705868769902628540, 461006369270166660143, 6919274132365824549936

Offset: 1

Vladeta Jovovic, Feb 14 2001

I. P. Goulden and D. M.Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

Georg Fischer, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (48,-932,9550,-56319,194762,-382908,387000,-151200).

Column k=6 of A059443.

Cf. A002718.

CoefficientList[Series[x^4*(16800*x^4-11362*x^3+2237*x^2-112*x-3) / ((1-x)*(2*x-1)*(3*x-1)*(4*x-1)*(6*x-1)*(7*x-1)*(10*x-1)*(15*x-1)), {x, 0, 21}], x] (* Georg Fischer, May 18 2019 *)

a(n)=(1/6!)*(15^n-6*10^n-15*7^n+30*6^n+60*4^n-50*3^n-180*2^n+240) \\ Georg Fischer, May 18 2019

a(n) = (1/6!)*(15^n - 6*10^n - 15*7^n + 30*6^n + 60*4^n - 50*3^n - 180*2^n + 240).
E.g.f.: exp(-x-1/2*x^2*(exp(y)-1)) * Sum_{i>=0} x^i/i!*exp(binomial(i, 2)*y), for m-block bicoverings of an n-set.
G.f.: x^4*(16800*x^4-11362*x^3+2237*x^2-112*x-3) / ((1-x)*(2*x-1)*(3*x-1)*(4*x-1)*(6*x-1)*(7*x-1)*(10*x-1)*(15*x-1)). [Colin Barker, Jan 11 2013; corrected by Georg Fischer, May 18 2019]

More terms from Colin Barker, Jan 11 2013

A059948 Number of 7-block bicoverings of an n-set.

Original entry on oeis.org

0, 0, 0, 0, 40, 3306, 131876, 3961356, 103290096, 2488179582, 57162274972, 1274774473632, 27887396866472, 602352276704178, 12899161619186388, 274612697648135028, 5822592730060070368, 123107330974129584294

Offset: 1

Vladeta Jovovic, Feb 14 2001

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Column k=7 of A059443.

Cf. A002718.

a(n)=(1/7!) * (21^n -7*15^n -21*11^n +42*10^n +105*7^n -140*6^n +105*5^n -420*4^n +35*3^n +1050*2^n -1050).
The number of m-block bicoverings of an n-set is [x^m*y^n] 1/n!*exp(-x-1/2*x^2*(exp(y)-1)) * sum(i>=0, x^i/i! * exp(binomial(i, 2)*y) ) where [x^m*y^n] extracts the coefficient of x^m*y^n, see Goulden/Jackson p.203.
G.f.: 2*x^5*(5197500*x^6-4601880*x^5+1501221*x^4-219455*x^3+12587*x^2+47*x-20) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(10*x-1)*(11*x-1)*(15*x-1)*(21*x-1)). - Colin Barker, Jul 07 2013

cp's OEIS Frontend

A060486 Tricoverings of an n-set.

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A060492 Triangle T(n,k) of k-block ordered tricoverings of an unlabeled n-set (n >= 3, k = 4..2n).

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A059530 Triangle T(n,k) of k-block T_0-tricoverings of an n-set, n >= 3, k = 0..2*n.

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

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A060491 Number of ordered tricoverings of an unlabeled n-set.

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

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Mathematica

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A060485 Number of 7-block tricoverings of an n-set.

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A059946 Number of 5-block bicoverings of an n-set.

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