A002718 -id:A002718 - 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

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

A178165 Number of unordered collections of distinct nonempty subsets of an n-element set where each element appears in at most 2 subsets.

Original entry on oeis.org

1, 2, 8, 59, 652, 9736, 186478, 4421018, 126317785, 4260664251, 166884941780, 7489637988545, 380861594219460, 21739310882945458, 1381634777325000263, 97089956842985393297, 7497783115765911443879, 632884743974716421132084

Offset: 0

Daniel E. Loeb, Dec 16 2010

nonn

If each element must appear in exactly 1 subset, then we get the Bell numbers A000110.

If each element must appear in exactly 2 subsets, then we get A002718.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Row n=2 of A330964.

Cf. A094574, A000110, A002718, A178171, A178173.

terms = m = 30;
a094577[n_] := Sum[Binomial[n, k]*BellB[2n-k], {k, 0, n}];
egf = Exp[(1 - Exp[x])/2]*Sum[a094577[n]*(x/2)^n/n!, {n, 0, m}] + O[x]^m;
A094574 = CoefficientList[egf + O[x]^m, x]*Range[0, m-1]!;
a[n_] := Sum[Binomial[n, k]*A094574[[k+1]], {k, 0, n}];
Table[a[n], {n, 0, m-1}] (* Jean-François Alcover, May 24 2019 *)

from numpy import array
def toBinary(n, k):
    ans=[]
    for i in range(k):
        ans.insert(0, n%2)
        n=n>>1
    return array(ans)
def powerSet(k): return [toBinary(n,k) for n in range(1,2**k)]
def courcelle(maxUses, remainingSets, exact=False):
    if exact and not all(maxUses<=sum(remainingSets)): ans=0
    elif len(remainingSets)==0: ans=1
    else:
        set0=remainingSets[0]
        if all(set0<=maxUses): ans=courcelle(maxUses-set0,remainingSets[1:],exact=exact)
        else: ans=0
        ans+=courcelle(maxUses,remainingSets[1:],exact=exact)
    return ans
for i in range(10):
    print(i, courcelle(array([2]*i),powerSet(i),exact=False))

Binomial transform of A094574: a(n) = Sum_{k=0..n} C(n,k)*A094574(k).

Edited and corrected by Max Alekseyev, Dec 19 2010

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

A098233 Consider the family of ordinary multigraphs. Sequence gives the triangle read by rows giving coefficients of polynomials arising from enumeration of those multigraphs on n edges.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 7, 3, 1, 1, 13, 46, 47, 25, 6, 1, 1, 40, 295, 587, 516, 235, 65, 10, 1, 1, 121, 1846, 6715, 9690, 7053, 3006, 800, 140, 15, 1, 1, 364, 11347, 73003, 170051, 189458, 119211, 46795, 12201, 2170, 266, 21, 1, 1, 1093, 68986, 768747

Offset: 0

N. J. A. Sloane, Oct 26 2004

Also gives number T(n, k) of partitions of the multiset {1, 1, 2, 2, ..., n, n} into k nonempty subsets, for 2 <= k <= 2n. - Marko Riedel, Jan 22 2023

			The first few polynomials are:
  1,
  x^2,
  x^2+x^3+x^4,
  x^2+4x^3+7x^4+3x^5+x^6,
  x^2+13x^3+46x^4+47x^5+25x^6+6x^7+x^8,
  x^2+40x^3+295x^4+587x^5+516x^6+235x^7+65x^8+10x^9+x^10,
  ...
Triangle starts:
  1;
  1;
  1,  1,   1;
  1,  4,   7,   3,   1;
  1, 13,  46,  47,  25,   6,  1;
  1, 40, 295, 587, 516, 235, 65, 10, 1;
  ...

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Steve Butler, Fan Chung, Jay Cummings, and R. L. Graham, Juggling card sequences, arXiv:1504.01426 [math.CO], 2015.
L. Comtet, Birecouvrements et birevêtements d'un ensemble fini, Studia Sci. Math. Hungar 3 (1968): 137-152. [Annotated scanned copy. Warning: the table of v(n,k) has errors.]
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

Cf. A360037, A360038, A360039, A020554 (row sums).

cp's OEIS Frontend

A060486 Tricoverings of an n-set.

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

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PARI

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

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PARI

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

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Mathematica

PARI

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

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A178165 Number of unordered collections of distinct nonempty subsets of an n-element set where each element appears in at most 2 subsets.

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Mathematica

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

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

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