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.

Showing 1-8 of 8 results.

A270227 Array read by antidiagonals: T(n,m) is the number of matchings in the rook graph K_n X K_m.

Original entry on oeis.org

1, 2, 2, 4, 7, 4, 10, 32, 32, 10, 26, 193, 370, 193, 26, 76, 1382, 5950, 5950, 1382, 76, 232, 11719, 122984, 270529, 122984, 11719, 232, 764, 112604, 3175696, 16873930, 16873930, 3175696, 112604, 764, 2620, 1221889, 98815588, 1384880065, 3337807996, 1384880065, 98815588, 1221889, 2620
Offset: 1

Views

Author

Andrew Howroyd, Mar 13 2016

Keywords

Comments

Observations: (for n+m <= 32)
Examination of values modulus a small prime yields several patterns.
T(n,m) == (n+1)*(m+1) (mod 2) for n+m>2.
T(n,m) == T(n,m+6) (mod 3).
T(n,m) is not divisible by 3.
T(n,m) == 0 (mod 5) for n==4 (mod 5) and m<>2 and except when m=n=4.
T(5,m) == 0 (mod 208) for m >= 13.
T(6,m) == 0 (mod 19) for m >= 19.

Examples

			The start of the sequence as table:
*   1     2       4         10           26              76 ...
*   2     7      32        193         1382           11719 ...
*   4    32     370       5950       122984         3175696 ...
*  10   193    5950     270529     16873930      1384880065 ...
*  26  1382  122984  168739305   3337807996    909046586596 ...
*  76 11719 3175696 1384880065 909046586596 855404716021831 ...
* ...

Links

Crossrefs

Main diagonal is A270228. Rows include A000085, A270229.

A270229 Number of matchings in the 2 X n rook graph P_2 X K_n.

Original entry on oeis.org

1, 2, 7, 32, 193, 1382, 11719, 112604, 1221889, 14639786, 192949639, 2760749048, 42732172993, 709490574158, 12596398359367, 237750425419508, 4757710386662401, 100516614496518866, 2236829315345704711, 52262526676903613264, 1279512810244450887361
Offset: 0

Views

Author

Andrew Howroyd, Mar 13 2016

Keywords

Comments

Sequence extended to n=0 using closed form. (binomial transform of A111883)

Links

Crossrefs

Cf. A270227, A270228, A000085, A081919 (perfect matchings).

Programs

  • Mathematica
    a[n_] := Sum[Binomial[n, k]*Abs[HermiteH[k, I/Sqrt[2]]]^2/2^k, {k, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 01 2017 *)
CoefficientList[Series[E^((2-x)*x/(1-x)) / Sqrt[1-x^2], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2017 *)

Formula

Binomial transform of A111883.
From Vaclav Kotesovec, Oct 01 2017: (Start)
a(n) = (n+1)*a(n-1) + (n-1)^2*a(n-2) - (n-2)*(n-1)^2*a(n-3) + (n-3)*(n-2)*(n-1)*a(n-4).
E.g.f.: exp((2-x)*x/(1-x)) / sqrt(1-x^2).
a(n) ~ exp(1/2 + 2*sqrt(n) - n) * n^n / 2.
(End)

A286422 Number of matchings in the n X n black bishop graph.

Original entry on oeis.org

2, 12, 130, 9492, 1166928, 1431128744, 2907639077764, 76670800431934272, 3341096345926174809912, 2311650738313947870105792416, 2645105778378736719464340469683304, 56641723029988800376624313271476598959936
Offset: 2

Views

Author

Andrew Howroyd, May 08 2017

Keywords

Comments

Matchings are not necessarily perfect matchings.
C# software that can be used to compute this sequence can be found in A270228.

Links

Crossrefs

Cf. A286423, A287248, A270228, A216078 (independent vertex sets), A234603 (cycles).

A286423 Number of matchings in the n X n white bishop graph.

Original entry on oeis.org

2, 7, 130, 4207, 1166928, 541240697, 2907639077764, 25886034363696809, 3341096345926174809912, 713738790204487208874991935, 2645105778378736719464340469683304, 16238808907214611705432043192158547965751
Offset: 2

Views

Author

Andrew Howroyd, May 08 2017

Keywords

Comments

Matchings are not necessarily perfect matchings.
C# software that can be used to compute this sequence can be found in A270228.

Links

Crossrefs

Cf. A286422, A287248, A270228, A216332 (independent vertex sets), A234630 (cycles).

A289198 Number of maximal matchings in the n X n rook graph.

Original entry on oeis.org

1, 2, 84, 22368, 84961440, 5429866337280, 7315512116927938560, 235781588994736418036121600, 207456452048917943576497565466624000, 5583211401338046269360238971594326671360000000, 5098207942457032504011606690585598401287135271321600000000
Offset: 1

Views

Author

Eric W. Weisstein, Jun 28 2017

Keywords

Links

Crossrefs

Main diagonal of A341847.
Cf. A270228, A281433, A289197, A292196.

Extensions

a(1) changed and a(5)-a(11) from Andrew Howroyd, Oct 05 2017

A289197 Number of maximum matchings in the n X n rook graph.

Original entry on oeis.org

1, 2, 72, 2016, 22615200, 61903180800, 383946391896883200, 291754347175555891200000, 1589674150230421751470227456000000, 653674192758917925444914420184514560000000, 4669210639701352189820773315967264103247065907200000000
Offset: 1

Views

Author

Eric W. Weisstein, Jun 28 2017

Keywords

Links

Crossrefs

Main diagonal of A341850.
Cf. A270228, A289198.

Extensions

Terms a(7) and beyond from Andrew Howroyd, Feb 21 2021

A270247 Number of matchings in the n X n torus grid graph C_n X C_n.

Original entry on oeis.org

1, 7, 370, 41025, 15637256, 23079663560, 127193770624285, 2645142169931308801, 206932904585998805434690, 60953421285412135689567940992, 67583556205239600880061198746186383, 282092296203355454009618109524478429807744
Offset: 1

Views

Author

Andrew Howroyd, Mar 13 2016

Keywords

Comments

C_{n} X C_{n} is also known as the (n,n)-torus grid graph.

Links

Crossrefs

Cf. A270246, A102080, A028420, A270228, A027683, A228316, A222199.

A293249 Number of matchings in the n-triangular honeycomb bishop graph.

Original entry on oeis.org

1, 3, 24, 624, 58258, 21423356, 33352210384, 234271975272172, 7849205797114197704, 1318509837007765330964948, 1161589241831987552884873549832, 5592410686612429649991093735982825800, 152814054664056604403587836700924195641924920
Offset: 1

Views

Author

Eric W. Weisstein, Oct 03 2017

Keywords

Comments

C# software that can be used to compute this sequence can be found in A270228 (with rook column heights of 1..n). - Andrew Howroyd, Oct 04 2017

Links

Crossrefs

Cf. A270228.

Extensions

a(8)-a(13) from Andrew Howroyd, Oct 04 2017
Showing 1-8 of 8 results.

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