A002501 -id:A002501 - cp's OEIS frontend

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

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).

A227322 Triangle read by rows: T(n, m) for 0 <= m <= n is the number of bipartite connected labeled graphs with parts of size n and m.

Original entry on oeis.org

1, 1, 1, 0, 1, 5, 0, 1, 19, 205, 0, 1, 65, 1795, 36317, 0, 1, 211, 14221, 636331, 23679901, 0, 1, 665, 106819, 10365005, 805351531, 56294206205, 0, 1, 2059, 778765, 162470155, 26175881341, 3735873535339, 502757743028605

Offset: 0

Pavel Irzhavski, Jul 06 2013

			Triangle T(n, m) begins:
n\m 0 1    2      3         4           5             6               7
0   1
1   1 1
2   0 1    5
3   0 1   19    205
4   0 1   65   1795     36317
5   0 1  211  14221    636331    23679901
6   0 1  665 106819  10365005   805351531   56294206205
7   0 1 2059 778765 162470155 26175881341 3735873535339 502757743028605
...
Consider labeled bipartite graph with parts of size 2 and 2. To make graph connected it is possible to use all four possible edges or omit any one of them. Thus T(2, 2) = 5.

F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68.

Cf. A226658, A123260.
Main diagonal gives: A005333.
Columns m=2, 3, 4 give: A001047, A002501, A002502.

T(n, m) = 2^(n*m) - sum for all (i, j) in ({1, 2, ..., n} X {1, 2, ..., m} UNION (1, 0)) \ (n, m) \ (1 - n, 0) of T(i, j)*C(n - 1, i - 1)*C(m, j)*2^((n - i)*(m - j)), where C(n, m) is the binomial coefficient (A007318). This relation can be obtained considering connected component which contains the first vertex of the largest part. (If the largest part has zero size we get T(0, 0) = 2^0 - 0 = 1 which is true.)

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

A092796 Number of connected relations.

Original entry on oeis.org

1, 213, 14857, 694485, 27005881, 957263493, 32333393737, 1064686990965, 34589700409561, 1115777278022373, 35856732186282217, 1149998292486777045, 36843831022923582841, 1179748027215029366853, 37764598757179830172297, 1208682260675932309564725

Offset: 1

Goran Kilibarda, Vladeta Jovovic, Apr 15 2004

G. C. Greubel, Table of n, a(n) for n = 1..660
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 (91,-3299,62713,-682172,4276972,-14386144,20106240).

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

[32^n - 5*17^n - 10*11^n + 20*10^n + 30*8^n - 60*7^n + 24*6^n: n in [0..50]]; // G. C. Greubel, Oct 08 2017

Table[32^n - 5*17^n - 10*11^n + 20*10^n + 30*8^n - 60*7^n + 24*6^n, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

for(n=0,50, print1(32^n - 5*17^n - 10*11^n + 20*10^n + 30*8^n - 60*7^n + 24*6^n, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = 32^n - 5*17^n - 10*11^n + 20*10^n + 30*8^n - 60*7^n + 24*6^n.

G.f.: -x*(132960*x^5 + 145292*x^4 - 17528*x^3 - 1227*x^2 + 122*x + 1) / ((6*x-1)*(7*x-1)*(8*x-1)*(10*x-1)*(11*x-1)*(17*x-1)*(32*x-1)). - Colin Barker, Jul 13 2013

Additional term from Colin Barker, Jul 13 2013

A092797 Number of connected relations.

Original entry on oeis.org

1, 667, 108817, 10796275, 858251401, 61283936827, 4147211888737, 273109341611395, 17736960725057401, 1143745441025278987, 73483870162431314257, 4712360023676936085715, 301901195708380781658601, 19331914197940256185117147, 1237580377249840094294765377

Offset: 1

Goran Kilibarda, Vladeta Jovovic, Apr 15 2004

G. C. Greubel, Table of n, a(n) for n = 1..550
G. Kilibarda and V. Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.

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

[64^n - 6*33^n - 15*19^n + 30*18^n - 10*15^n + 120*12^n - 120*11^n  + 30*10^n - 270*9^n + 360*8^n - 120*7^n: n in [0..50]]; // G. C. Greubel, Oct 08 2017

Table[64^n - 6*33^n - 15*19^n + 30*18^n - 10*15^n + 120*12^n - 120*11^n  + 30*10^n - 270*9^n + 360*8^n - 120*7^n, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

for(n=0,50, print1(64^n - 6*33^n - 15*19^n + 30*18^n - 10*15^n + 120*12^n - 120*11^n  + 30*10^n - 270*9^n + 360*8^n - 120*7^n, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = 64^n - 6*33^n - 15*19^n + 30*18^n - 10*15^n + 120*12^n - 120*11^n + 30*10^n - 270*9^n + 360*8^n - 120*7^n.

G.f.: x*(54888451200*x^9 +55706052240*x^8 -14450714964*x^7 -624924*x^6 +247511131*x^5 -22659769*x^4 +564934*x^3 +10694*x^2 -461*x -1) / ((7*x -1)*(8*x -1)*(9*x -1)*(10*x -1)*(11*x -1)*(12*x -1)*(15*x -1)*(18*x -1)*(19*x -1)*(33*x -1)*(64*x -1)). - Colin Barker, Jul 13 2013

More terms from Colin Barker, Jul 13 2013

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

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A227322 Triangle read by rows: T(n, m) for 0 <= m <= n is the number of bipartite connected labeled graphs with parts of size n and m.

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A092794 Number of connected relations.

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A092795 Number of connected relations.

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A092796 Number of connected relations.

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A092797 Number of connected relations.

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