A093732 -id:A093732

A002501

a(n) = 7^n - 3*4^n + 2*3^n.
(Formerly M5078 N2197)

+10
17

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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."

REFERENCES

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).

LINKS

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)

FORMULA

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

A diagonal of A262307.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

A002502

Number of connected relations.
(Formerly M5336 N2323)

+10
17

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

REFERENCES

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).

LINKS

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).

FORMULA

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

MATHEMATICA

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

CROSSREFS

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

A diagonal of A262307.

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

A093733

Number of connected relations.

+10
13

1, 665, 106819, 10365005, 805351531, 56294206205, 3735873535339, 241600284318365, 15423235216318411, 978180744322139645, 61834480769377286059, 3902270609960140639325, 246057483524862034206091, 15508484277325946034039485, 977254123876968508188975979

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

T. D. Noe, Table of n, a(n) for n = 1..100

Goran Kilibarda and Vladeta 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 (-195, 15886, -726290, 20952193, -403792115, 5336718048, -48588590600, 299693200656, -1195947048240, 2785165036416, -2872859996160).

FORMULA

a(n) = 63^n - 6*32^n - 15*18^n + 30*17^n - 10*14^n + 120*11^n - 120*10^n + 30*9^n - 270*8^n + 360*7^n - 120*6^n.

G.f.: x*(96368590080*x^9 + 27682953984*x^8 - 13185435000*x^7 + 774468980*x^6 + 143028190*x^5 - 19071533*x^4 + 626800*x^3 + 6970*x^2 - 470*x - 1) / ((6*x -1)*(7*x -1)*(8*x -1)*(9*x -1)*(10*x -1)*(11*x -1)*(14*x -1)*(17*x -1)*(18*x -1)*(32*x -1)*(63*x -1)). - Colin Barker, Jul 07 2013

MATHEMATICA

Table[63^n-6*32^n-15*18^n+30*17^n-10*14^n+120*11^n-120*10^n+30*9^n-270*8^n+360*7^n-120*6^n, {n, 1, 25}] (* G. C. Greubel, Oct 06 2017 *)

CoefficientList[Series[x (96368590080x^9+27682953984x^8-13185435000x^7+774468980x^6+ 143028190x^5-19071533x^4+626800x^3+6970x^2-470x-1)/((6x-1)(7x-1)(8x-1)(9x-1)(10x-1)(11x-1)(14x-1)(17x-1)(18x-1)(32x-1)(63x-1)), {x, 0, 20}], x] (* or *) LinearRecurrence[{195, -15886, 726290, -20952193, 403792115, -5336718048, 48588590600, -299693200656, 1195947048240, -2785165036416, 2872859996160}, {0, 1, 665, 106819, 10365005, 805351531, 56294206205, 3735873535339, 241600284318365, 15423235216318411, 978180744322139645}, 20] (* Harvey P. Dale, Sep 23 2023 *)

PROG

(PARI) for(n=1, 25, print1(63^n-6*32^n-15*18^n+30*17^n-10*14^n+120*11^n-120*10^n+30*9^n-270*8^n+360*7^n-120*6^n, ", ")) \\ G. C. Greubel, Oct 06 2017

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Goran Kilibarda and Vladeta Jovovic, Apr 14 2004

STATUS

approved

A114936

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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.

FORMULA

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).

MATHEMATICA

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 *)

PROG

(PARI) 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

CROSSREFS

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

KEYWORD

easy,nonn

AUTHOR

Goran Kilibarda and Vladeta Jovovic, Jan 08 2006

STATUS

approved

A114935

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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.

FORMULA

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

MATHEMATICA

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 *)

PROG

(PARI) 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

CROSSREFS

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

KEYWORD

easy,nonn

AUTHOR

Goran Kilibarda and Vladeta Jovovic, Jan 08 2006

STATUS

approved

A114937

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..16.

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

FORMULA

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).

CROSSREFS

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

KEYWORD

easy,nonn

AUTHOR

Goran Kilibarda and Vladeta Jovovic, Jan 08 2006

STATUS

approved

A114934

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

+10
4

0, 0, 0, 21, 2773, 148365, 5878391, 204819447, 6721694469, 214306917321, 6736603947907, 210284186632443, 6541309609120385, 203129541349695597, 6302428271530970943, 195459285517696665759, 6060542952694406463421

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..670

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

FORMULA

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) - 30*exp(6*x) - 96*exp(5*x) - 45*exp(4*x) + 20*exp(3*x) + 50*exp(2*x) + 24*exp(x) - 24).

MATHEMATICA

With[{nmax = 50}, CoefficientList[Series[(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] - 30*Exp[6*x] - 96*Exp[5*x] - 45*Exp[4*x] + 20*Exp[3*x] + 50*Exp[2*x] + 24*Exp[x] - 24), {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 07 2017 *)

PROG

(PARI) x='x+O('x^50); concat([0, 0, 0], Vec(serlaplace((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) - 30*exp(6*x) - 96*exp(5*x) - 45*exp(4*x) + 20*exp(3*x) + 50*exp(2*x) + 24*exp(x) - 24)))) \\ G. C. Greubel, Oct 07 2017

CROSSREFS

Cf. A002501, A002502, A093732, A093732, A114933, A114935, A114936, A114937.

KEYWORD

easy,nonn

AUTHOR

Goran Kilibarda and Vladeta Jovovic, Jan 08 2006

STATUS

approved

A114933

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

+10
3

0, 0, 0, 32, 1094, 23055, 405475, 6575842, 102567444, 1569195485, 23775369725, 358461659952, 5391042181294, 80974624209115, 1215462744452775, 18238484835400862, 273628186560143144, 4104820038944901945

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

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.

FORMULA

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

MATHEMATICA

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] + 12*Exp[4*x] - Exp[3*x] - 11*Exp[2*x] - 6*Exp[x] + 6), {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 07 2017 *)

PROG

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

CROSSREFS

Cf. A002501, A002502, A093732, A093732, A114934, A114935, A114936, A114937.

KEYWORD

easy,nonn

AUTHOR

Goran Kilibarda and Vladeta Jovovic, Jan 08 2006

STATUS

approved

A226658

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

+10
3

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

EXAMPLE

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

CROSSREFS

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)

KEYWORD

nonn,tabl

AUTHOR

R. H. Hardin Jun 14 2013

STATUS

approved

A114932

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

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.

FORMULA

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

MATHEMATICA

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 *)

PROG

(PARI) 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

CROSSREFS

Cf. A002501, A002502, A093732, A093733.

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

KEYWORD

easy,nonn

AUTHOR

Goran Kilibarda and Vladeta Jovovic, Jan 08 2006

STATUS

approved