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 1-3 of 3 results.

A303913 Array read by antidiagonals: T(n,k) is the number of (planar) unlabeled asymmetric k-ary cacti having n polygons.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 3, 2, 0, 1, 1, 0, 6, 10, 8, 0, 1, 1, 0, 10, 28, 54, 18, 0, 1, 1, 0, 15, 60, 193, 222, 61, 0, 1, 1, 0, 21, 110, 505, 1140, 1107, 170, 0, 1, 1, 0, 28, 182, 1095, 3876, 7688, 5346, 538, 0, 1, 1, 0, 36, 280, 2093, 10326, 33125, 52364, 27399, 1654, 0
Offset: 0

Views

Author

Andrew Howroyd, May 02 2018

Keywords

Comments

A k-ary cactus is a planar k-gonal cactus with vertices on each polygon numbered 1..k counterclockwise with shared vertices having the same number. In total there are always exactly k ways to number a given cactus since all polygons are connected. See the reference for a precise definition. - Andrew Howroyd, Feb 18 2020

Examples

			Array begins:
===============================================================
n\k| 1   2     3      4       5        6        7         8
---+-----------------------------------------------------------
0  | 1   1     1      1       1        1        1         1 ...
1  | 1   1     1      1       1        1        1         1 ...
2  | 0   0     0      0       0        0        0         0 ...
3  | 0   1     3      6      10       15       21        28 ...
4  | 0   2    10     28      60      110      182       280 ...
5  | 0   8    54    193     505     1095     2093      3654 ...
6  | 0  18   222   1140    3876    10326    23394     47208 ...
7  | 0  61  1107   7688   33125   107056   285383    662620 ...
8  | 0 170  5346  52364  290700  1149126  3621150   9702008 ...
9  | 0 538 27399 373560 2661100 12845166 47813367 147765409 ...
...

Links

Crossrefs

Columns k=2..7 are A054358, A054422, A052395, A054364, A054367, A054370.
Cf. A303694, A303912.

Programs

  • Mathematica
    T[0, _] = 1;
T[n_, k_] := DivisorSum[n, MoebiusMu[n/#] Binomial[k #, #] &]/n - (k-1) Binomial[n k, n]/((k-1) n + 1);
Table[T[n-k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, May 22 2018 *)
  • PARI
    T(n,k)={if(n==0, 1, sumdiv(n, d, moebius(n/d)*binomial(k*d, d))/n - (k-1)*binomial(k*n, n)/((k-1)*n+1))}

Formula

T(n,k) = (Sum_{d|n} mu(n/d)*binomial(k*d, d))/n - (k-1)*binomial(k*n, n)/((k-1)*n+1) for n > 0.

A054362 Number of unlabeled 4-gonal cacti having n polygons.

Original entry on oeis.org

1, 1, 1, 3, 11, 52, 307, 1936, 13207, 93496, 683988, 5127163, 39230669, 305299420, 2410624122, 19273255184, 155780437711, 1271253542364, 10462650241996, 86765190816362, 724450039738076, 6086167189623746, 51416796881915019
Offset: 0

Views

Author

Simon Plouffe

Keywords

Comments

Also, the number of noncrossing partitions up to rotation composed of n blocks of size 4. - Andrew Howroyd, Apr 30 2018

Links

Crossrefs

Column k=4 of A303694.
Cf. A052394, A052395, A303870.

Programs

  • Maple
    with(combinat): with(numtheory): m := 4: for p from 2 to 28 do s1 := 0: s2 := 0: for d from 1 to p do if p mod d = 0 then s1 := s1+phi(p/d)*binomial(m*d, d) fi: od: for d from 1 to p-1 do if gcd(m, p-1) mod d = 0 then s2 := s2+phi(d)*binomial((p*m)/d, (p-1)/d) fi: od: printf(`%d, `, (s1+s2)/(m*p)-binomial(m*p, p)/(p*(m-1)+1)) od: # Zerinvary Lajos, Dec 01 2006
  • Mathematica
    a[0] = 1;
a[n_] := (DivisorSum[n, EulerPhi[n/#] Binomial[4#, #]&] + DivisorSum[GCD[n - 1, 4], EulerPhi[#] Binomial[4n/#, (n-1)/#]&])/(4n) - Binomial[4n, n]/ (3n + 1);
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Jun 29 2018, after Andrew Howroyd *)
  • PARI
    a(n) = {if(n==0, 1, (sumdiv(n, d, eulerphi(n/d)*binomial(4*d, d)) + sumdiv(gcd(n-1, 4), d, eulerphi(d)*binomial(4*n/d, (n-1)/d)))/(4*n) - binomial(4*n, n)/(3*n+1))} \\ Andrew Howroyd, Apr 30 2018

Formula

a(n) = ((Sum_{d|n} phi(n/d)*binomial(4*d, d)) + (Sum_{d|gcd(n-1, 4)} phi(d)*binomial(4*n/d, (n-1)/d)))/(4*n) - binomial(4*n, n)/(3*n+1) for n > 0. - Andrew Howroyd, Apr 30 2018

Extensions

More terms from Zerinvary Lajos, Dec 01 2006

A052394 Number of unlabeled 4-ary cacti having n polygons.

Original entry on oeis.org

1, 1, 4, 10, 44, 197, 1228, 7692, 52828, 373636, 2735952, 20506258, 156922676, 1221179926, 9642496488, 77092885016, 623121750844, 5085013101164, 41850600967984, 347060754685888, 2897800158952304, 24344668688424333, 205667187527660076, 1746375819789491996, 14898241072028602276
Offset: 0

Views

Author

Simon Plouffe

Keywords

Links

Crossrefs

Column k=4 of A303912.
Cf. A054362, A052395.

Programs

  • Mathematica
    a[n_] := If[n == 0, 1, (Binomial[4*n, n]/(3*n + 1) + DivisorSum[n, Binomial[4*#, #]*EulerPhi[n/#]*Boole[# < n] & ])/n]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Jul 17 2017 *)
  • PARI
    a(n) = if(n==0, 1, sumdiv(n, d, eulerphi(n/d)*binomial(4*d, d))/n - 3*binomial(4*n, n)/(3*n+1)) \\ Andrew Howroyd, May 02 2018

Formula

a(n) = (1/n)*(Sum_{d|n} phi(n/d)*binomial(4*d, d)) - 3*binomial(4*n, n)/(3*n+1) for n > 0. - Andrew Howroyd, May 02 2018
a(n) ~ 2^(8*n + 1/2) / (sqrt(Pi) * n^(5/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jul 17 2017

Extensions

More terms from Jean-François Alcover, Jul 17 2017
Showing 1-3 of 3 results.

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