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.

User: Andy Hardt

Andy Hardt's wiki page.

Andy Hardt has authored 7 sequences.

A218090 Number of unlabeled point-determining bipartite graphs on n vertices.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 8, 17, 63, 224, 1248, 8218, 75992, 906635, 14447433, 303100595, 8415834690, 309390830222, 15105805368214, 982300491033887
Offset: 0

Views

Author

Andy Hardt, Oct 20 2012

Keywords

Comments

A graph is point-determining if no two vertices have the same set of neighbors. This kind of graph is also called a mating graph.

Examples

			Consider n = 3. The triangle graph is point-determining, but it is not bipartite, so it is not counted in a(3). The graph *--*--* is bipartite, but it is not point-determining (the vertices on the two ends have the same neighborhood), so it is also not counted in a(3). The only graph counted in a(3) is the graph *--*  *. - _Justin M. Troyka_, Nov 27 2013

Links

Crossrefs

Cf. A006024, A004110 (labeled and unlabeled point-determining graphs).
Cf. A092430, A004108 (labeled and unlabeled connected point-determining graphs).
Cf. A232699 (labeled point-determining bipartite graphs).
Cf. A232700, A088974 (labeled and unlabeled connected point-determining bipartite graphs).

A193777 Number of signed permutations of size 2n invariant under D and D'bar and avoiding (-2, 1) and (2, -1).

Original entry on oeis.org

1, 2, 6, 18, 58, 190, 642, 2206, 7746, 27662, 100738, 373550, 1413506, 5457710, 21546466, 87025806, 360264258, 1529624366, 6669850466, 29877013902, 137560725890, 650780790894, 3162711095074, 15774862353614, 80687636530882, 422713072650286, 2265833731786594
Offset: 0

Views

Author

Andy Hardt, Aug 04 2011

Keywords

Comments

Also the number of signed permutations of size n invariant under D and avoiding (-2, 1) and (2, -1).

Links

Crossrefs

Cf. A193778, A000085.

Programs

  • Maple
    inv := proc(n) option remember; if n<2 then 1 else inv(n-1)+(n-1)*inv(n-2) fi end:
a := proc(n) option remember; if n < 2 then n+1 else
2*a(n-1) + n*a(n-2) - add(j*a(j)*inv(n-j-2), j=1..n-3) fi end:
seq(a(n), n=0..26); # Peter Luschny, Dec 09 2018
  • Mathematica
    inv[n_] := inv[n] = If[n<2, 1, inv[n-1] + (n-1) inv[n-2]];
a[n_] := a[n] = If[n<2, n+1, 2 a[n-1] + n a[n-2] - Sum[j a[j] inv[n-j-2], {j, 1, n-3}]];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jun 26 2019, after Peter Luschny *)

Formula

a(n) = 2*a(n-1) + n*a(n-2) - Sum_{j=1..k-3} j*a(j)*|S_{k-j-3}^D|, where S_n^D is the set of unsigned permutations of length n invariant under D.
a(n) = 2*a(n-1) + n*a(n-2) - Sum_{j=1..n-3} j*a(j)*A000085(n-j-2). - Andrew Howroyd, Dec 09 2018

Extensions

Terms a(9) and beyond from Peter Luschny, Dec 09 2018

A193778 Number of signed permutations of length 2n invariant under D and D'bar.

Original entry on oeis.org

1, 2, 8, 32, 160, 832, 4864, 29696, 195584, 1341440, 9723904, 73105408, 574062592, 4657184768, 39165624320, 339133595648, 3028204650496, 27760959422464, 261439835078656, 2521668748574720, 24912764963127296, 251559029812232192, 2595790316527157248
Offset: 0

Views

Author

Andy Hardt, Aug 04 2011

Keywords

Comments

See section 6 in the Hardt reference.

Examples

			For n = 1, the permutations of length 2 are (-1, 2) and (1, -2).

Links

Crossrefs

Column k=4 of A376826.
Cf. A193777, A000085.

Programs

  • Maple
    exp(2*x*(x + 1)): series(%, x, 23):
seq(n!*coeff(%, x, n), n = 0..22); # Peter Luschny, May 30 2021
  • Mathematica
    RecurrenceTable[{a[0]==1,a[1]==2,a[n]==2a[n-1]+4(n-1)a[n-2]},a,{n,30}] (* Harvey P. Dale, Mar 14 2021 *)
a[n_] := 2^n HypergeometricPFQ[{-n/2, (1-n)/2}, {}, 2];
Table[a[n], {n, 0, 22}] (* Peter Luschny, May 30 2021 *)
  • PARI
    seq(n)={my(v=vector(n+1)); v[1]=1; v[2]=2; for(n=2, n, v[n+1] = 2*v[n] + 4*(n-1)*v[n-1]); v} \\ Andrew Howroyd, Dec 08 2018

Formula

a(n) = 2*a(n-1) + 4*(n - 1)*a(n-2).
From Peter Luschny, May 30 2021: (Start)
a(n) = 2^n * hypergeom([-n/2, (1 - n)/2], [], 2).
a(n) = 2^n * A000085(n).
a(n) = n! * [x^n] exp(2*x*(x + 1)). (End)

Extensions

Terms a(6) and beyond from Andrew Howroyd, Dec 08 2018

A193776 Number of signed permutations of length n invariant under the reverse complement and avoiding (-2, -1), (-2, 1), (2, -1).

Original entry on oeis.org

1, 2, 3, 5, 12, 17, 65, 80, 473, 527, 4444, 4679, 51391, 52628, 703659, 711449, 11098896, 11156477, 197809793, 198299024, 3927270089, 3931960343, 85908742132, 85958728847, 2052375195679, 2052960568556, 53160174898371, 53167638586121, 1483752628890840, 1483855482962885, 44391655829672177
Offset: 0

Views

Author

Andy Hardt, Aug 04 2011

Keywords

Examples

			For n = 2, the permutations are (1, 2), (2, 1), (-1, -2), (-2, -1).

Programs

  • Mathematica
    a[n_] := a[n] = If[EvenQ[n],
   Sum[a[2j]*(n/2 - j - 1)!, {j, 0, n/2 - 1}] + 2^(n/2)*(n/2)!,
   Sum[a[2j+1]*((1/2)*(n - 2j - 3))!, {j, 0, (n - 3)/2}] +
      (2^((n - 1)/2) + 1)*((n - 1)/2)!];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 10 2024 *)

Formula

a(2k) = 2^k k! + \sum_{j=0}^{k-1}(k-j-1)! a(2j)
a(2k+1) = (2^k + 1)k! + \sum_{j=0}^{k-1}(k-j-1)! a(2j+1)

A193775 Number of signed permutations invariant under the reverse complement and avoiding (-2, -1), (-1, 2), and (1, -2).

Original entry on oeis.org

1, 2, 3, 4, 10, 12, 52, 58, 394, 418
Offset: 0

Views

Author

Andy Hardt, Aug 04 2011

Keywords

Examples

			For n = 3, the permutations are (1, 2, 3), (3, 2, 1), (-1, -2, -3), and (-3, 2, -1).

Formula

a(n) = (\lfloor n/2\rfloor)! 2^{\lfloor n/2\rfloor} + \sum_{k=0}^{\lfloor n/2\lceil} k!

A193774 Number of signed permutations of length n invariant under the reverse complement and avoiding (-2, 1) and (2, -1).

Original entry on oeis.org

1, 2, 4, 6, 20, 26, 124, 146, 956, 1050, 9100
Offset: 0

Views

Author

Andy Hardt, Aug 04 2011

Keywords

Examples

			For n = 2, the permutations are (1, 2), (2, 1), (-1, -2), and (-2, -1).

A193763 Number of signed permutations of length n avoiding (-2, 1) and (2, -1).

Original entry on oeis.org

1, 2, 6, 22, 94, 462, 2606, 16862, 124782, 1048990, 9921550, 104447550, 1211190638, 15329157278, 210093682254, 3097760346238, 48869022535726, 821007386273118, 14630266558195214, 275575669958063678, 5469996402416702958, 114107289124208861470
Offset: 0

Views

Author

Andy Hardt, Aug 04 2011

Keywords

Comments

Also the number of signed permutations of length 2*n invariant under Dbar and avoiding (-1, 2) and (1, -2).
Also the number of signed permutations of length 2*n invariant under R180bar and avoiding (-1, 2) and (1, -2).

Examples

			For n = 2, the 6 permutations are (2, 1), (-2, -1), (1, 2), (1, -2), (-1, 2), and (-1, -2).
a(3) = 22 = sum of top row terms of M^3 = (11 + 3 + 2 + 6); where 11 = A051296(3).

Crossrefs

Cf. A051296.

Programs

  • Maple
    b := proc(n) option remember; if n = 0 then 2 else
add(factorial(k)*b(n-k), k=1..n) fi end:
a := n -> if n = 0 then return 1 else b(n) end:
seq(a(n), n=0..21); # Peter Luschny, Dec 07 2018

Formula

a(0) = 1, and for n > 0, a(n) = n! + Sum_{j=0..n-1} (n-j)! * a(j).
a(n) is the sum of top row terms of M^n, M = an infinite square production matrix as follows:
1, 1, 0, 0, 0, 0,...
2, 0, 2, 0, 0, 0,...
3, 0, 0, 3, 0, 0,...
4, 0, 0, 0, 4, 0,...
5, 0, 0, 0, 0, 5,...
... The upper left term of M^n = A051296(n). - Gary W. Adamson, Sep 26 2011

Extensions

More terms from Joerg Arndt, Aug 16 2011

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