cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-30 of 45 results. Next

A323292 Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have two vertices in common.

Original entry on oeis.org

1, 0, 0, 1, 0, 15, 160, 4125, 193200, 19384225
Offset: 0

Views

Author

Gus Wiseman, Jan 10 2019

Keywords

Examples

			The a(5) = 15 hypergraphs:
  {{1,2,3},{1,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{3,4,5}}
  {{1,2,5},{1,3,4}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{3,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,3,4},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,4,5},{2,3,4}}
  {{1,4,5},{2,3,5}}
Non-isomorphic representatives of the 3 unlabeled 3-uniform hypergraphs spanning 6 vertices where no two edges have two vertices in common, and their multiplicities in the labeled case which add up to a(6) = 160:
   10 X {{1,2,3},{4,5,6}}
  120 X {{1,3,5},{2,3,6},{4,5,6}}
   30 X {{1,2,4},{1,3,5},{2,3,6},{4,5,6}}

Crossrefs

Cf. A000665, A025035, A125791, A190865, A289837, A302374, A302394, A320395, A322451, A323293-A323299.

Programs

  • Mathematica
    stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]>=2&],Union@@#==Range[n]&]],{n,6}]

Formula

Inverse binomial transform of A323293. - Andrew Howroyd, Aug 14 2019

Extensions

a(9) from Andrew Howroyd, Aug 14 2019

A323294 Number of 3-uniform hypergraphs spanning n labeled vertices where every two edges have two vertices in common.

Original entry on oeis.org

1, 0, 0, 1, 11, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431
Offset: 0

Views

Author

Gus Wiseman, Jan 10 2019

Keywords

Examples

			The a(4) = 11 hypergraphs:
  {{1,2,3},{1,2,4}}
  {{1,2,3},{1,3,4}}
  {{1,2,3},{2,3,4}}
  {{1,2,4},{1,3,4}}
  {{1,2,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4}}
  {{1,2,3},{1,2,4},{2,3,4}}
  {{1,2,3},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

Links

Crossrefs

Cf. A000217, A000665, A025035, A125791, A161680, A289837, A302374, A302394, A319540, A320395, A322451, A323292-A323299.

Programs

  • Mathematica
    stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]<=1&],Union@@#==Range[n]&]],{n,10}]
  • PARI
    seq(n)={Vec(serlaplace(1 - x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x + O(x^(n-1)))/2))} \\ Andrew Howroyd, Aug 18 2019

Formula

a(n) = binomial(n,2) for n >= 5. - Gus Wiseman, Jan 16 2019
Binomial transform is A289837. - Gus Wiseman, Jan 16 2019
a(n) = A000217(n-1) for n >= 5. - Alois P. Heinz, Jan 24 2019
E.g.f.: 1 - x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x)/2. - Andrew Howroyd, Aug 18 2019

A323297 Number of 3-uniform hypergraphs on n labeled vertices where no two edges have exactly one vertex in common.

Original entry on oeis.org

1, 1, 1, 2, 16, 76, 271, 1212, 10158, 78290, 503231, 3495966, 33016534, 327625520, 3000119669, 28185006956, 308636238516, 3631959615948, 42031903439809, 493129893459310, 6264992355842706, 84639308481270656, 1159506969481515271, 16131054826385628592
Offset: 0

Views

Author

Gus Wiseman, Jan 11 2019

Keywords

Examples

			The a(4) = 16 hypergraphs:
  {}
  {{1,2,3}}
  {{1,2,4}}
  {{1,3,4}}
  {{2,3,4}}
  {{1,2,3},{1,2,4}}
  {{1,2,3},{1,3,4}}
  {{1,2,3},{2,3,4}}
  {{1,2,4},{1,3,4}}
  {{1,2,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4}}
  {{1,2,3},{1,2,4},{2,3,4}}
  {{1,2,3},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
The following are non-isomorphic representatives of the 8 unlabeled 3-uniform hypergraphs on 6 vertices with no two edges having exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(6) = 271:
   1 X {}
  20 X {{1,2,3}}
  90 X {{1,3,4},{2,3,4}}
  10 X {{1,2,3},{4,5,6}}
  60 X {{1,4,5},{2,4,5},{3,4,5}}
  60 X {{1,2,4},{1,3,4},{2,3,4}}
  15 X {{1,5,6},{2,5,6},{3,5,6},{4,5,6}}
  15 X {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

Links

Crossrefs

Cf. A000665, A025035, A125791, A289837, A299471, A302374, A302394, A306021, A323293, A323294, A323296.

Programs

  • Mathematica
    stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]==1&]],{n,8}]
  • PARI
    seq(n)={Vec(serlaplace(exp(x - x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x + O(x^(n-1)))/2)))} \\ Andrew Howroyd, Aug 18 2019

Formula

Binomial transform of A323296.
E.g.f.: exp(x - x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x)/2). - Andrew Howroyd, Aug 18 2019

Extensions

a(10)-a(11) from Alois P. Heinz, Aug 11 2019
Terms a(12) and beyond from Andrew Howroyd, Aug 18 2019

A323298 Number of 3-uniform hypergraphs spanning n labeled vertices where every two edges have exactly one vertex in common.

Original entry on oeis.org

1, 0, 0, 1, 0, 15, 150, 1815, 0, 945, 0, 10395, 0, 135135, 0, 2027025, 0, 34459425, 0, 654729075, 0, 13749310575, 0, 316234143225, 0, 7905853580625, 0, 213458046676875, 0, 6190283353629375, 0, 191898783962510625, 0, 6332659870762850625, 0, 221643095476699771875
Offset: 0

Views

Author

Gus Wiseman, Jan 11 2019

Keywords

Comments

The only way to cover more than 7 vertices is with edges all having a single common vertex. For the special cases of n = 6 or n = 7, there are also covers without a common vertex. - Andrew Howroyd, Aug 15 2019

Examples

			The a(5) = 15 hypergraphs:
  {{1,4,5},{2,3,5}}
  {{1,4,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,4},{2,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,2,5},{3,4,5}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{1,3,4}}
  {{1,2,4},{3,4,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{1,4,5}}
The following are non-isomorphic representatives of the 5 unlabeled 3-uniform hypergraphs spanning 7 vertices in which every two edges have exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(7) = 1815.
  105 X {{1,2,7},{3,4,7},{5,6,7}}
  840 X {{1,4,5},{2,4,6},{3,4,7},{5,6,7}}
  630 X {{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
  210 X {{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
   30 X {{1,2,7},{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
From _Andrew Howroyd_, Aug 15 2019: (Start)
The following are non-isomorphic representatives of the 2 unlabeled 3-uniform hypergraphs spanning 6 vertices in which every two edges have exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(6) = 150.
    120 X {{1,2,3},{1,4,5},{3,5,6}}
     30 X {{1,2,3},{1,4,5},{3,5,6},{2,4,6}}
(End)

Links

Crossrefs

Cf. A001147, A025035, A125791, A190865, A289837, A299471, A302374, A302394, A322451, A323293, A323296, A323297, A323299.

Programs

  • Mathematica
    stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]!=1&],Union@@#==Range[n]&]],{n,10}]
  • PARI
    a(n)={if(n%2, if(n<=3, n==3, if(n==7, 1815, n!/(2^(n\2)*(n\2)!))), if(n==6, 150, n==0))} \\ Andrew Howroyd, Aug 15 2019

Formula

a(2*n) = 0 for n > 3; a(2*n-1) = A001147(n) for n > 4. - Andrew Howroyd, Aug 15 2019

Extensions

Terms a(13) and beyond from Andrew Howroyd, Aug 15 2019

A370357 Number of partitions of [3n] into n sets of size 3 avoiding any set {3j-2,3j-1,3j} (1<=j<=n).

Original entry on oeis.org

1, 0, 9, 252, 14337, 1327104, 182407545, 34906943196, 8877242235393, 2896378850249568, 1179516253790272041, 586470881874514605660, 349649630741370155550849, 246214807676005971547223712, 202182156277565590613022234777, 191496746966087534845272710637564
Offset: 0

Views

Author

Alois P. Heinz, Feb 16 2024

Keywords

Examples

			a(0) = 1: the empty partition satisfies the condition.
a(1) = 0: 123 is not counted.
a(2) = 9: 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234 are counted. 123|456 is not counted.

Links

Crossrefs

Column k=0 of A370347.
Column k=3 of A370366.
Cf. A000007, A025035, A059841, A370358.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<3, [1, 0, 9][n+1],
      9*(n*(n-1)/2*a(n-1)+(n-1)^2*a(n-2)+(n-1)*(n-2)/2*a(n-3)))
    end:
seq(a(n), n=0..20);

Formula

a(n) = Sum_{j=0..n} (-1)^(n-j) * binomial(n,j) * A025035(j).
a(n) = A025035(n) - A370358(n).
a(n) mod 9 = A000007(n).
a(n) mod 2 = A059841(n).

A370358 Number of partitions of [3n] into n sets of size 3 having at least one set {3j-2,3j-1,3j} (1<=j<=n).

Original entry on oeis.org

0, 1, 1, 28, 1063, 74296, 8182855, 1305232804, 284438292607, 81167321350432, 29367491879327959, 13135455977606994340, 7116140280642196449151, 4591529352468711908776288, 3479040085783649820897765223, 3058744793640846605215609362436
Offset: 0

Views

Author

Alois P. Heinz, Feb 16 2024

Keywords

Examples

			a(1) = 1: 123.
a(2) = 1: 123|456.
a(3) = 28: 123|456|789, 123|457|689, 123|458|679, 123|459|678, 123|467|589, 123|468|579, 123|469|578, 123|478|569, 123|479|568, 123|489|567, 124|356|789, 125|346|789, 126|345|789, 127|389|456, 128|379|456, 129|378|456, 134|256|789, 135|246|789, 136|245|789, 137|289|456, 138|279|456, 139|278|456, 145|236|789, 146|235|789, 156|234|789, 178|239|456, 179|238|456, 189|237|456.

Links

Crossrefs

Cf. A025035, A057427, A059841, A370347, A370357.
Column k=3 of A370363.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n<3, [1, 0, 9][n+1],
      9*(n*(n-1)/2*b(n-1)+(n-1)^2*b(n-2)+(n-1)*(n-2)/2*b(n-3)))
    end:
a:= n-> (3*n)!/(n!*(3!)^n)-b(n):
seq(a(n), n=0..20);

Formula

a(n) = Sum_{j=0..n-1} (-1)^(n-j+1) * binomial(n,j) * A025035(j).
a(n) = A025035(n) - A370357(n).
a(n) = Sum_{k=1..n} A370347(n,k).
a(n) mod 2 = A059841(n) for n>=2.
a(n) mod 9 = A057427(n).

A135429 Number of transitive reflexive binary relations R on n labeled elements where |{y : xRy}| <= 3 for all x.

Original entry on oeis.org

1, 1, 4, 29, 210, 2116, 25522, 362832, 6000276, 113593688, 2434603356, 58523364604, 1565365441708, 46273309903536, 1502773485741816, 53336787604185656, 2059209704215556448, 86117458019804680576, 3886421648246467359364, 188615552477984650605744
Offset: 0

Views

Author

Alois P. Heinz, Dec 12 2007

Keywords

References

  • A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Links

Crossrefs

Cf. A135312, A025035, A006882, A008277, A000142.

Programs

  • Maple
    A025035:= proc(n) option remember; (3*n)! /n! /(6^n) end:
z:= proc(n) option remember; add(binomial(n,k+k) *doublefactorial(k+k-1) *k^(n-k-k), k=0..floor(n/2)) end:
r:= proc(n) option remember; n! * add(add(add(add(Stirling2(e,d) *a^(d+i) *(a*(a+1)/2)^(n-i-i-e-d-a) /a! /(n-i-i-e-d-a)! /i! /e! /(2^i), a=0..(n-i-i-e-d)), d=0..min(e,n-i-i-e)), e=0..(n-i-i)), i=0..floor(n/2)) end:
a:= proc(n) option remember; n! *add(add(A025035(i) *z(j) *r(n-3*i-j) /(3*i)! /j! /(n-3*i-j)!, j=0..(n-3*i)), i=0..floor(n/3)) end:
seq(a(n), n=0..30);
  • Mathematica
    Unprotect[Power]; 0^0 = 1; A025035[n_] := A025035[n] = (3n)!/n!/6^n; z[n_] := z[n] = Sum[Binomial[n, k+k]*(k+k-1)!!*k^(n-k-k), {k, 0, Floor[n/2]}]; r[n_] := r[n] = n!*Sum[Sum[Sum[Sum[StirlingS2[e, d]*a^(d+i)*(a*(a+1)/2)^(n-i-i-e-d-a)/a!/(n-i-i-e-d-a)!/i!/e!/2^i, {a, 0, n-i-i-e-d}], {d, 0, Min[e, n-i-i-e]}], {e, 0, n-i-i}], {i, 0, Floor[n/2]}]; a[n_] := a[n] = n!*Sum[Sum[A025035[i]*z[j]*r[n-3*i-j]/(3i)!/j!/(n-3*i-j)!, {j, 0, n-3*i}], {i, 0, Floor[n/3]}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz *)

Formula

a(n) = see program.

A200313 E.g.f. satisfies: A(x) = exp(x^3*A(x)^3/3!).

Original entry on oeis.org

1, 1, 70, 28000, 33833800, 91842150400, 471920698849600, 4105733038511104000, 55918460253906250000000, 1124922893768186370457600000, 31962429471680921191680301600000, 1237813985055170041194334820761600000, 63474917512551971525535771981021376000000
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x^3/3! + 70*x^6/6! + 28000*x^9/9! + 33833800*x^12/12! + ...
where log(A(x)) = x^3*A(x)^3/3! and
A(x)^3 = 1 + 3*x^3/3! + 270*x^6/6! + 120960*x^9/9! + 155925000*x^12/12! + ...

Links

Crossrefs

Cf. A025035, A034941, A200314, A200315.

Programs

  • GAP
    List([0..10],n->(3*n+1)^(n-1)*Factorial(3*n)/(Factorial(n)*Factorial(3)^n)); # Muniru A Asiru, Jul 28 2018
  • Magma
    [(3*n+1)^(n-1)*Factorial(3*n)/(6^n*Factorial(n)): n in [0..30]]; // G. C. Greubel, Jul 27 2018
  • Mathematica
    Table[(3*n + 1)^(n - 1)*(3*n)!/(n!*(3!)^n), {n, 0, 30}] (* G. C. Greubel, Jul 27 2018 *)
  • PARI
    {a(n)=(3*n)!*polcoeff(1/x*serreverse(x*(exp(-x^3/3!+x*O(x^(3*n))))),3*n)}
  • PARI
    {a(n)=(3*n+1)^(n-1)*(3*n)!/(n!*(3!)^n)}

Formula

a(n) = (3*n+1)^(n-1) * (3*n)!/(n!*(3!)^n).
E.g.f.: (1/x)*Series_Reversion( x*exp(-x^3/3!) ).
Powers of e.g.f.: define a(n,m) by A(x)^m = Sum_{n>=0} a(n,m)*x^(3*n)/(3*n)!
then a(n,m) = m*(3*n+m)^(n-1) * (3*n)!/(n!*(3!)^n).

A326587 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 3 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 11, 10, 0, 645, 924, 280, 0, 111563, 197802, 101640, 15400, 0, 42567981, 86271640, 57717660, 15415400, 1401400, 0, 30342678923, 67630651098, 53492240256, 19158419280, 3144741600, 190590400
Offset: 0

Views

Author

Peter Luschny, Jul 20 2019

Keywords

Examples

			Triangle starts:
0 [1]
1 [0, 1]
2 [0, 11, 10]
3 [0, 645, 924, 280]
4 [0, 111563, 197802, 101640, 15400]
5 [0, 42567981, 86271640, 57717660, 15415400, 1401400]
6 [0, 30342678923, 67630651098, 53492240256, 19158419280, 3144741600, 190590400]

Crossrefs

Row sums A243664. Main diagonal A025035.
A129062 (m=1, associated with A131689), A326477 (m=2, associated with A241171), this sequence (m=3, associated with A278073), A326585 (m=4, associated with A278074).

Programs

Formula

T(n, k) = T_{3}(n, k) where T_{m}(n, k) is defined in A326477.

A334250 Number of set partitions of [3n] into 3-element subsets {i, i+k, i+2k} with 1<=k<=n.

Original entry on oeis.org

1, 1, 2, 4, 12, 35, 129, 567, 2920, 16110, 103467, 717608, 5748214, 47937957, 441139750, 4319093093, 45963368076, 510202534002, 6150655137844, 76789781005325, 1028853084775725, 14294680087131380
Offset: 0

Views

Author

Alois P. Heinz, Apr 20 2020

Keywords

Comments

Differs from A331621 first at n=7.

Examples

			a(2) = 2: 123|456, 135|246.
a(3) = 4: 123|456|789, 123|468|579, 135|246|789, 147|258|369.

Links

Crossrefs

Cf. A014307 (the same for 2-element subsets), A025035, A059108, A104429 (where k is not restricted), A285527, A331621, A337520.
Main diagonal of A360334.

Programs

  • Maple
    b:= proc(s, t) option remember; `if`(s={}, 1, (m-> add(
     `if`({m-j, m-2*j} minus s={}, b(s minus {m, m-j, m-2*j},
            t), 0), j=1..min(t, iquo(m-1, 2))))(max(s)))
    end:
a:= proc(n) option remember; forget(b): b({$1..3*n}, n) end:
seq(a(n), n=0..12);
  • Mathematica
    b[s_List, t_] := b[s, t] = If[s == {}, 1, Function[m, Sum[If[{m - j, m - 2j} ~Complement~ s == {}, b[s ~Complement~ {m, m - j, m - 2j}, t], 0], {j, 1, Min[t, Quotient[m - 1, 2]]}]][Max[s]]];
a[n_] := a[n] = b[Range[3n], n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 12}] (* Jean-François Alcover, May 10 2020, after Maple *)

Formula

a(n) <= A104429(n) <= A025035(n).

Extensions

a(17)-a(21) from Martin Fuller, Jul 19 2025
Previous Showing 21-30 of 45 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.