A054557 -id:A054557 - cp's OEIS frontend

A054559 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 8 1-simplexes.

30, 180, 630, 1680, 3780, 7560, 13860, 23760, 38610, 60060, 90090, 131040, 185640, 257040, 348840, 465120, 610470, 790020, 1009470, 1275120, 1593900, 1973400, 2421900, 2948400, 3562650, 4275180, 5097330, 6041280, 7120080, 8347680, 9738960, 11309760, 13076910

Offset: 5

Vladeta Jovovic, Apr 10 2000

Number of {T_1,T_2,...,T_k} where T_i,i=1..k are 3-subsets of an n-set such that {D | D is 2-subset of T_i for some i=1..k} has l elements; k=5,l=8.

Let H be the n X n Hilbert matrix H(i,j) = 1/(i+j-1) for 1 <= i,j <= n. Let B be the inverse matrix of H. The sum of the elements in row 3 of B equals -a(n+2). - T. D. Noe, May 01 2011

V. Jovovic, On the number of two-dimensional simplicial complexes (in Russian), Metody i sistemy tekhnicheskoy diagnostiki, Vypusk 16, Mezhvuzovskiy zbornik nauchnykh trudov, Izdatelstvo Saratovskogo universiteta, 1991.

Vincenzo Librandi, Table of n, a(n) for n = 5..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A054557.

Cf. A000389, A052787.

I:=[30, 180, 630, 1680, 3780, 7560]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..30]]; // Vincenzo Librandi, Apr 29 2012

Table[n*(n+1)*(n+2)*(n+3)*(n+4)/4, {n,1,100}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2009 *)
CoefficientList[Series[30/(1-x)^6,{x,0,30}],x] (* Vincenzo Librandi, Apr 29 2012 *)

x='x+O('x^30); Vec(serlaplace(x^5*exp(x)/4)) \\ G. C. Greubel, Nov 23 2017

a(n) = 30*C(n,5) = 30*A000389(n) = n*(n-1)*(n-2)*(n-3)*(n-4)/4.
G.f.: 30*x^5/(1-x)^6. - Colin Barker, Jan 19 2012
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Vincenzo Librandi, Apr 29 2012
E.g.f.: x^5*exp(x)/4. - G. C. Greubel, Nov 23 2017
From Amiram Eldar, Mar 08 2022: (Start)
Sum_{n>=5} 1/a(n) = 1/24.
Sum_{n>=5} (-1)^(n+1)/a(n) = 8*log(2)/3 - 131/72. (End)

A054647 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 4 2-simplexes and 12 1-simplexes.

Original entry on oeis.org

30, 2310, 42840, 391545, 2375100, 10980585, 41761720, 136963255, 399689290, 1060984925, 2603641040, 5979294230, 12973080120, 26794003110, 53000811600, 100914240770, 185718969590, 331524753560, 575738427880, 975199600375, 1614655942900, 2618302433175

Offset: 6

Vladeta Jovovic, Apr 16 2000

Number of {T_1,T_2,...,T_k} where T_i,i=1..k are 3-subsets of an n-set such that {D | D is 2-subset of T_i for some i=1..k} has l elements; k=4,l=12.

Numbers of sets of 4 triangles that are pairwise edge-disjoint in the complete graph K_n. - Julian Allagan, Mar 08 2025

Julian Allagan, Edge-Disjoint Triangle Packings in Complete Graphs: Recurrence Relations and Closed Formulas (submitted 2025)
V. Jovovic, On the number of two-dimensional simplicial complexes (in Russian), Metody i sistemy tekhnicheskoy diagnostiki, Vypusk 16, Mezhvuzovskiy zbornik nauchnykh trudov, Izdatelstvo Saratovskogo universiteta, 1991.

T. D. Noe, Table of n, a(n) for n = 6..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A054557, A054558, A054559, A054560, A054561, A054562, A381862, A381863.

a(n) = 30*C(n, 6)+2100*C(n, 7)+25200*C(n, 8)+86625*C(n, 9)+116550*C(n, 10)+69300*C(n, 11)+15400*C(n, 12) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n^6+3*n^5-86*n^4-240*n^3+2704*n^2+5232*n-34128)/31104.

G.f.: 5*x^6*(169*x^6-1119*x^5+2535*x^4-1245*x^3-3030*x^2-384*x-6)/(x-1)^13. [Colin Barker, Jun 22 2012]

More terms from James Sellers, Apr 16 2000

A054558 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 9 1-simplexes.

Original entry on oeis.org

150, 960, 3605, 10360, 25200, 54600, 108570, 201960, 356070, 600600, 975975, 1536080, 2351440, 3512880, 5135700, 7364400, 10377990, 14395920, 19684665, 26565000, 35420000, 46703800, 60951150, 78787800, 100941750, 128255400

Offset: 5

Vladeta Jovovic, Apr 10 2000

Number of {T_1,T_2,...,T_k} where T_i, i=1..k are 3-subsets of an n-set such that {D | D is 2-subset of T_i for some i=1..k} has l elements; k=5, l=9.

V. Jovovic, On the number of two-dimensional simplicial complexes (in Russian), Metody i sistemy tekhnicheskoy diagnostiki, Vypusk 16, Mezhvuzovskiy zbornik nauchnykh trudov, Izdatelstvo Saratovskogo universiteta, 1991.

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A054557.

A054558:=n->n*(n-1)*(n-2)*(n-3)*(n-4)*(n^2+n+150)/144; seq(A054558(n), n=5..30); # Wesley Ivan Hurt, Apr 29 2014

Table[n*(n - 1)*(n - 2)*(n - 3)*(n - 4)*(n^2 + n + 150)/144, {n, 5, 30}] (* Wesley Ivan Hurt, Apr 29 2014 *)

a(n) = 150*C(n,5) +60*C(n,6) +35*C(n,7) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n^2+n+150)/144.

G.f.: 5*x^5*(30-48*x+25*x^2)/(1-x)^8. - Colin Barker, Jun 21 2012

A054562 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 13 1-simplexes.

Original entry on oeis.org

540, 65625, 1272320

Offset: 6

Vladeta Jovovic, Apr 10 2000

Number of {T_1,T_2,...,T_k} where T_i,i=1..k are 3-subsets of an n-set such that {D | D is 2-subset of T_i for some i=1..k} has l elements; k=5,l=13.

V. Jovovic, On the number of two-dimensional simplicial complexes (in Russian), Metody i sistemy tekhnicheskoy diagnostiki, Vypusk 16, Mezhvuzovskiy zbornik nauchnykh trudov, Izdatelstvo Saratovskogo universiteta, 1991.

Cf. A054557.

a(n)=540*C(n, 6)+61845*C(n, 7)+762440*C(n, 8)+...

A054560 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 11 1-simplexes.

Original entry on oeis.org

6300, 89586, 549528

Offset: 6

Vladeta Jovovic, Apr 10 2000

Number of {T_1,T_2,...,T_k} where T_i,i=1..k are 3-subsets of an n-set such that {D | D is 2-subset of T_i for some i=1..k} has l elements; k=5,l=11.

V. Jovovic, On the number of two-dimensional simplicial complexes (in Russian), Metody i sistemy tekhnicheskoy diagnostiki, Vypusk 16, Mezhvuzovskiy zbornik nauchnykh trudov, Izdatelstvo Saratovskogo universiteta, 1991.

Cf. A054557.

a(n)=6300*C(n, 6)+45486*C(n, 7)+9240*C(n, 8)+...

A054561 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 12 1-simplexes.

Original entry on oeis.org

2700, 118020, 1220520

Offset: 6

Vladeta Jovovic, Apr 10 2000

Number of {T_1,T_2,...,T_k} where T_i,i=1..k are 3-subsets of an n-set such that {D | D is 2-subset of T_i for some i=1..k} has l elements; k=5,l=12.

V. Jovovic, On the number of two-dimensional simplicial complexes (in Russian), Metody i sistemy tekhnicheskoy diagnostiki, Vypusk 16, Mezhvuzovskiy zbornik nauchnykh trudov, Izdatelstvo Saratovskogo universiteta, 1991.

Cf. A054557.

a(n)=2700*C(n, 6)+99120*C(n, 7)+351960*C(n, 8)+...

A054648 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 4 2-simplexes and 11 1-simplexes.

Original entry on oeis.org

360, 13230, 137760, 835380, 3679200, 13056120, 39584160, 106383420, 259819560, 586936350, 1242521280, 2489618040, 4758324480, 8728907040, 15446635200, 26477304840, 44114190120, 71649152190, 113722852320, 176771479500, 269590120800, 404035889400, 595897192800

Offset: 6

Vladeta Jovovic, Apr 16 2000

nonn

Number of {T_1,T_2,...,T_k} where T_i,i=1..k are 3-subsets of an n-set such that {D | D is 2-subset of T_i for some i=1..k} has l elements; k=4,l=11.

V. Jovovic, On the number of two-dimensional simplicial complexes (in Russian), Metody i sistemy tekhnicheskoy diagnostiki, Vypusk 16, Mezhvuzovskiy zbornik nauchnykh trudov, Izdatelstvo Saratovskogo universiteta, 1991.

T. D. Noe, Table of n, a(n) for n = 6..1000

Cf. A054557-A054562.

a(n) = 360*C(n, 6)+10710*C(n, 7)+42000*C(n, 8)+41580*C(n, 9)+12600*C(n, 10) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n^4+3*n^3-58*n^2-120*n+1008)/288.

Empirical G.f.: -30*x^6*(89*x^4-391*x^3+401*x^2+309*x+12)/(x-1)^11. [Colin Barker, Jun 22 2012]

More terms from James Sellers, Apr 16 2000

cp's OEIS Frontend

A054559 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 8 1-simplexes.

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A054647 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 4 2-simplexes and 12 1-simplexes.

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A054558 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 9 1-simplexes.

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A054562 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 13 1-simplexes.

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A054560 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 11 1-simplexes.

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A054561 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 12 1-simplexes.

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A054648 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 4 2-simplexes and 11 1-simplexes.

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