A093733 -id:A093733 - cp's OEIS frontend

A002501 a(n) = 7^n - 34^n + 23^n.

1, 19, 205, 1795, 14221, 106819, 778765, 5581315, 39606541, 279447619, 1965098125, 13792018435, 96690872461, 677427332419, 4744368982285, 33220131761155, 232579232659981, 1628208214321219, 11398072876175245, 79788974736297475, 558532690864457101

Offset: 1

N. J. A. Sloane

Counts connected relations. On page 578 Kreweras (1969) says: "Le théorème s'applique notamment au dénombrement des relations binaires externes qui possèdent la propriété de connexité; cela revient à calculer le nombre a(m,n) de manières de remplir un tableau de m lignes et n colonnes avec des 0 et des 1, en respectant les deux conditions suivantes: (1): aucune rangée (ligne ni colonne) ne doit être tout entière remplie de zéros; (2): deux cases quelconques marquées 1 peuvent être jointes par une chaîne de cases marquées 1 telle que deux cases consécutives de la chaîne appartiennent à une même rangée."

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..200
G. Kreweras, Inversion des polynomes de Bell bidimensionnels et application au dénombrement des relations binaires connexes, C. R. Acad. Sci. Paris Ser. A-B 268 1969 A577-A579.
Index entries for linear recurrences with constant coefficients, signature (14,-61,84)

Cf. A005333, A001047, A002502, A093732, A093733.

A diagonal of A262307.

Table[7^n - 3*4^n + 2*3^n, {n, 20}] (* T. D. Noe, May 29 2012 *)

a(n)=7^n-3*4^n+2*3^n \\ Charles R Greathouse IV, Sep 24 2015

G.f.: -x*(1+5*x) / ( (3*x-1)*(7*x-1)*(4*x-1) ). - R. J. Mathar, Jun 09 2013

a(n) = 14*a(n-1) - 61*a(n-2) + 84*a(n-3). - Wesley Ivan Hurt, Apr 11 2022

Better definition and more terms from Goran Kilibarda, Vladeta Jovovic, Apr 14 2004

A002502 Number of connected relations.

Original entry on oeis.org

1, 65, 1795, 36317, 636331, 10365005, 162470155, 2495037197, 37898120011, 572284920845, 8614868501515, 129467758660877, 1943971108806091, 29175170378428685, 437752102106036875, 6567275797761209357

Offset: 1

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..100
G. Kreweras, Inversion des polynômes de Bell bidimensionnels et application au dénombrement des relations binaires connexes. C. R. Acad. Sci. Paris Ser. A-B 268 1969 A577-A579.
Index entries for linear recurrences with constant coefficients, signature (38,-539,3622,-11640,14400).

Cf. A005333, A001047, A002501, A093732, A093733.

A diagonal of A262307.

LinearRecurrence[{38,-539,3622,-11640,14400},{1,65,1795,36317,636331},20] (* Harvey P. Dale, Mar 24 2017 *)

15^n-4*8^n-3*6^n+12*5^n-6*4^n. - Goran Kilibarda, Vladeta Jovovic, Apr 14 2004
G.f. x*( -1-27*x+136*x^2+480*x^3 ) / ( (6*x-1)*(5*x-1)*(15*x-1)*(4*x-1)*(8*x-1) ).
- R. J. Mathar, Jun 09 2013

More terms from Goran Kilibarda, Vladeta Jovovic, Apr 14 2004

A093732 Number of connected relations.

Original entry on oeis.org

1, 211, 14221, 636331, 23679901, 805351531, 26175881341, 831358677451, 26094426008221, 814105545191851, 25320182311228861, 786251347986776971, 24394981288950302941, 756583120577782494571, 23459491617092461686781, 727330825918603925122891

Offset: 1

Goran Kilibarda, Vladeta Jovovic, Apr 14 2004

T. D. Noe, Table of n, a(n) for n = 1..200
G. Kilibarda and V. Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.
G. Kreweras, Inversion des polynômes de Bell bidimensionnels et application au dénombrement des relations binaires connexes, C. R. Acad. Sci. Paris Ser. A-B 268 1969 A577-A579.
Index entries for linear recurrences with constant coefficients, signature (84,-2774,47548,-462525,2575088,-7643820,9374400).

Cf. A005333, A001047, A002501, A002502, A093733.

Table[31^n - 5*16^n - 10*10^n + 20*9^n + 30*7^n - 60*6^n + 24*5^n, {n, 25}] (* T. D. Noe, May 29 2012 *)

a(n)=31^n-5*16^n-10*10^n+20*9^n+30*7^n-60*6^n+24*5^n \\ Charles R Greathouse IV, Jun 16 2015

a(n) = 31^n - 5*16^n - 10*10^n + 20*9^n + 30*7^n - 60*6^n + 24*5^n.

G.f.: -x*(1+127*x-729*x^2-20467*x^3+107048*x^4+259620*x^5) / ( (9*x-1)*(6*x-1)*(7*x-1)*(5*x-1)*(31*x-1)*(10*x-1)*(16*x-1) ). - R. J. Mathar, Jun 09 2013

A114936 Number of connected (4,n)-hypergraphs (without empty edges).

Original entry on oeis.org

0, 1, 10, 135, 1992, 30166, 458885, 6965225, 105358102, 1588998756, 23915093535, 359444209015, 5397938190512, 81022969645346, 1215801458118985, 18240857019892005, 273644796626023722, 4104936328561231936

Offset: 0

Goran Kilibarda and Vladeta Jovovic, Jan 08 2006

G. C. Greubel, Table of n, a(n) for n = 0..845
Goran Kilibarda and Vladeta Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.

Cf. A002501, A002502, A093732, A093732, A093733, A093734, A114935, A114937.

With[{nmax = 50}, CoefficientList[Series[(1/4!)*(Exp[15*x] - 4*Exp[8*x] + 6*Exp[7*x] - 3*Exp[6*x] + 12*Exp[5*x] - 24*Exp[4*x] + 23*Exp[3*x] - 11*Exp[2*x] + 6*Exp[x] - 6), {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 07 2017 *)

x='x+O('x^50); concat([0], Vec(serlaplace((1/4!)*(exp(15*x) - 4*exp(8*x) + 6*exp(7*x) - 3*exp(6*x) + 12*exp(5*x) - 24*exp(4*x) + 23*exp(3*x) - 11*exp(2*x) + 6*exp(x) - 6)))) \\ G. C. Greubel, Oct 07 2017

E.g.f.: (1/4!)*(exp(15*x) - 4*exp(8*x) + 6*exp(7*x) - 3*exp(6*x) + 12*exp(5*x) - 24*exp(4*x) + 23*exp(3*x) - 11*exp(2*x) + 6*exp(x) - 6).

A114935 Number of connected (3,n)-hypergraphs (without empty edges).

Original entry on oeis.org

0, 1, 6, 44, 332, 2476, 18136, 130824, 933372, 6610676, 46603616, 327603904, 2298933412, 16115938476, 112906938696, 790735321784, 5536710117452, 38763269947876, 271368229299376, 1899679393564464, 13298164198917492

Offset: 0

Goran Kilibarda and Vladeta Jovovic, Jan 08 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000
Goran Kilibarda and Vladeta Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.

Cf. A002501, A002502, A093732, A093732, A093733, A114934, A114936, A114937.

With[{nmax = 50}, CoefficientList[Series[(1/3!)*(Exp[7*x] - 3*Exp[4*x] + 5*Exp[3*x] - 3*Exp[2*x] + 2*Exp[x] - 2), {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 07 2017 *)

x='x+O('x^50); concat([0], Vec(serlaplace((1/3!)*(exp(7*x) -3*exp(4*x) +5*exp(3*x) -3*exp(2*x) +2*exp(x) - 2)))) \\ G. C. Greubel, Oct 07 2017

E.g.f.: (1/3!)*(exp(7*x) -3*exp(4*x) +5*exp(3*x) -3*exp(2*x) +2*exp(x) - 2).

A114937 Number of connected (5,n)-hypergraphs (without empty edges).

Original entry on oeis.org

0, 1, 15, 336, 8880, 254596, 7606446, 231899522, 7137539256, 220623286632, 6831984816402, 211719998195278, 6562887569336652, 203453536535818388, 6307290799931347878, 195532244201392935354, 6061637498660735815968

Offset: 0

Goran Kilibarda and Vladeta Jovovic, Jan 08 2006

Goran Kilibarda and Vladeta Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.

Cf. A002501, A002502, A093732, A093733, A114932-A114936.

E.g.f.: (1/5!)*(exp(31*x) - 5*exp(16*x) + 10*exp(15*x) - 10*exp(10*x) + 20*exp(9*x) - 40*exp(8*x) + 65*exp(7*x) - 90*exp(6*x) + 144*exp(5*x) - 165*exp(4*x) + 120*exp(3*x) - 50*exp(2*x) + 24*exp(x) - 24).

A226658 T(n,k)=Number of nXk 0..4 arrays of sums of 2X2 subblocks of some (n+1)X(k+1) binary array.

Original entry on oeis.org

5, 19, 19, 65, 205, 65, 211, 1795, 1795, 211, 665, 14221, 36317, 14221, 665, 2059, 106819, 636331, 636331, 106819, 2059, 6305, 778765, 10365005, 23679901, 10365005, 778765, 6305, 19171, 5581315, 162470155, 805351531, 805351531, 162470155

Offset: 1

R. H. Hardin Jun 14 2013

Table starts
......5.........19...........65............211.............665............2059
.....19........205.........1795..........14221..........106819..........778765
.....65.......1795........36317.........636331........10365005.......162470155
....211......14221.......636331.......23679901.......805351531.....26175881341
....665.....106819.....10365005......805351531.....56294206205...3735873535339
...2059.....778765....162470155....26175881341...3735873535339.502757743028605
...6305....5581315...2495037197...831358677451.241600284318365
..19171...39606541..37898120011.26094426008221
..58025..279447619.572284920845
.175099.1965098125
.527345

			Some solutions for n=3 k=4
..1..1..1..0....0..1..1..1....0..1..1..0....1..0..1..1....1..0..2..3
..3..1..1..2....1..1..2..1....1..2..2..2....1..2..3..2....1..2..2..1
..3..2..2..3....3..1..2..3....2..1..2..3....2..3..2..1....3..3..1..0

R. H. Hardin, Table of n, a(n) for n = 1..71

Diagonal is A005333(n+1)
Staircase diagonal is A123281(n-3)
Column 1 is A001047(n+1)
Column 2 is A002501(n+1)
Column 3 is A002502(n+1)
Column 4 is A093732(n+1)
Column 5 is A093733(n+1)

A114932 Number of connected (3,n)-hypergraphs (without empty edges and without multiple edges).

Original entry on oeis.org

0, 0, 1, 25, 267, 2265, 17471, 128765, 927067, 6591505, 46545591, 327428805, 2298406067, 16114352345, 112902172111, 790721005645, 5536667136267, 38763140938785, 271367842141031, 1899678231827285, 13298160713181667

Offset: 0

Goran Kilibarda and Vladeta Jovovic, Jan 08 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000
Goran Kilibarda and Vladeta Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.

Cf. A002501, A002502, A093732, A093733.

Cf. A114933, A114934, A114935, A114936, A114937.

With[{nmax = 50}, CoefficientList[Series[(1/3!)*(Exp[7*x] - 3*Exp[4*x] - Exp[3*x] + 3*Exp[2*x] + 2*Exp[x] - 2), {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 07 2017 *)

x='x+O('x^50); concat([0,0], Vec(serlaplace((1/3!)*(exp(7*x)-3*exp(4*x)-exp(3*x)+3*exp(2*x)+2*exp(x)-2)))) \\ G. C. Greubel, Oct 07 2017

E.g.f.: (1/3!)*(exp(7*x)-3*exp(4*x)-exp(3*x)+3*exp(2*x)+2*exp(x)-2).

A092794 Number of connected relations.

Original entry on oeis.org

1, 21, 265, 2733, 25441, 223461, 1895545, 15736413, 128882641, 1046542101, 8451838825, 68020609293, 546227922241, 4380272835141, 35094966838105, 281025802973373, 2249545355064241, 18003091856638581, 144058517372685385, 1152637601335180653

Offset: 1

Goran Kilibarda, Vladeta Jovovic, Apr 15 2004

G. C. Greubel, Table of n, a(n) for n = 1..1000
G. Kilibarda and V. Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (17,-92,160).

Cf. A005333, A001047, A002501, A002502, A093732, A093733.

CoefficientList[Series[-x*(4*x + 1)/((4*x - 1)*(5*x - 1)*(8*x - 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 05 2017 *)

x='x+O('x^50); Vec(x*(4*x+1)/((1-4*x)*(1-5*x)*(1-8*x))) \\ G. C. Greubel, Oct 05 2017

a(n) = 8^n - 3*5^n + 2*4^n.
From Colin Barker, Jul 13 2013: (Start)
a(n) = 17*a(n-1) - 92*a(n-2) + 160*a(n-3).
G.f.: x*(4*x+1) / ((1-4*x)*(1-5*x)*(1-8*x)). (End)

Additional term from Colin Barker, Jul 13 2013

A092795 Number of connected relations.

Original entry on oeis.org

1, 67, 1993, 43891, 836521, 14764627, 249723433, 4123297651, 67157947561, 1085384064787, 17464790421673, 280328391247411, 4493290901135401, 71964955947764947, 1152089156508284713, 18439265231953981171, 295080697103288816041, 4721762414918959913107

Offset: 1

Goran Kilibarda, Vladeta Jovovic, Apr 15 2004

G. C. Greubel, Table of n, a(n) for n = 1..825
G. Kilibarda and V. Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (43,-701,5477,-20658,30240).

Cf. A005333, A001047, A002501, A002502, A093732, A093733.

[16^n - 4*9^n - 3*7^n + 12*6^n - 6*5^n: n in [1..50]]; // G. C. Greubel, Oct 08 2017

Table[16^n - 4*9^n - 3*7^n + 12*6^n - 6*5^n, {n, 1, 50}] (* G. C. Greubel, Oct 08 2017 *)
LinearRecurrence[{43,-701,5477,-20658,30240},{1,67,1993,43891,836521},20] (* Harvey P. Dale, May 24 2025 *)

for(n=1,50, print1(16^n - 4*9^n - 3*7^n + 12*6^n - 6*5^n, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = 16^n - 4*9^n - 3*7^n + 12*6^n - 6*5^n.

G.f.: x*(318*x^3+187*x^2-24*x-1) / ((5*x-1)*(6*x-1)*(7*x-1)*(9*x-1)*(16*x-1)). - Colin Barker, Jul 13 2013

More terms from Colin Barker, Jul 13 2013

cp's OEIS Frontend

A002501 a(n) = 7^n - 3*4^n + 2*3^n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A002502 Number of connected relations.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A093732 Number of connected relations.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A114936 Number of connected (4,n)-hypergraphs (without empty edges).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A114935 Number of connected (3,n)-hypergraphs (without empty edges).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A114937 Number of connected (5,n)-hypergraphs (without empty edges).

Views

Author

Keywords

Links

Crossrefs

Formula

A226658 T(n,k)=Number of nXk 0..4 arrays of sums of 2X2 subblocks of some (n+1)X(k+1) binary array.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A114932 Number of connected (3,n)-hypergraphs (without empty edges and without multiple edges).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A002501 a(n) = 7^n - 34^n + 23^n.