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.
Salah Uddin Mohammad's wiki page.
Salah Uddin Mohammad has authored 11 sequences. Here are the ten most recent ones:
For example, all the posets up to 3 elements are Riordan posets.
Triangle begins:
1;
2, 1;
5, 3, 1;
16, 9, 4, 1;
52, 31, 14, 5, 1;
188, 108, 52, 20, 6, 1;
690, 402, 193, 80, 27, 7, 1;
2638, 1523, 744, 315, 116, 35, 8, 1;
10272, 5934, 2908, 1261, 483, 161, 44, 9, 1;
40782, 23505, 11580, 5085, 2010, 707, 216, 54, 10, 1;
The connected posets counted in the first three rows of the triangle are shown by using the Hasse diagram as follows:
-------
o
|
o
--------------------------
| o
o o o | |
/ \ \ / | o
o o o | |
| o
----------------------------------------------------------
o o o o o o | |
/|\ \|/ |X| | | o
o o o o o o | o o o o | |
| | \ / / \ | o
o o | o o o o | |
| / \ | / \ | \ / | o
o o o \ | o o o o | |
\ / | \ | | o
o o o | |
Triangle begins:
1;
1, 1;
2, 1, 1;
2, 2, 1, 1;
4, 3, 2, 1, 1;
5, 4, 3, 2, 1, 1;
9, 6, 5, 3, 2, 1, 1;
13, 10, 6, 5, 3, 2, 1, 1;
23, 15, 10, 7, 5, 3, 2, 1, 1;
35, 24, 14, 10, 7, 5, 3, 2, 1, 1;
61, 39, 23, 14, 11, 7, 5, 3, 2, 1, 1;
98, 63, 34, 21, 14, 11, 7, 5, 3, 2, 1, 1;
In particular, the rooted trees counted in the first three rows of the triangle are shown by using the Hasse diagram as follows:
---------
o o
\ /
o
----------------------
o |
| |
o o | o o o
\ / | \ | /
o | o
------------------------------------------------------
o o o o | o |
\ / | | | | |
o o o o | o o o | o o o o
\ / \ / | \ | / | \ \ / /
o o | o | o
Triangle begins:
1;
1, 1;
3, 1, 1;
9, 4, 1, 1;
31, 12, 4, 1, 1;
115, 46, 13, 4, 1, 1;
474, 173, 49, 13, 4, 1, 1;
2097, 727, 188, 50, 13, 4, 1, 1;
9967, 3195, 795, 191, 50, 13, 4, 1, 1;
50315, 15017, 3502, 810, 192, 50, 13, 4, 1, 1;
...
Triangle begins:
1;
1, 1;
3, 1, 1;
9, 4, 1, 1;
30, 12, 4, 1, 1;
103, 45, 13, 4, 1, 1;
375, 160, 48, 13, 4, 1, 1;
1400, 613, 175, 49, 13, 4, 1, 1;
5380, 2354, 680, 178, 49, 13, 4, 1, 1;
...
Triangle begins:
1;
1, 1;
3, 1, 1;
7, 4, 1, 1;
15, 10, 4, 1, 1;
31, 28, 11, 4, 1, 1;
63, 67, 31, 11, 4, 1, 1;
127, 167, 80, 32, 11, 4, 1, 1;
...
B(x) = x*(1 - 2*x + 2*x^2)/((1 - x)*(1 - 2*x))
T(n)=[Vecrev(p/y) | p<-Vec(-1 + exp(sum(k=1, n, y^k*B(x^k)/k + O(x*x^n))))]
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 13 2022
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
max(1, 2^(d-1)-1), d=numtheory[divisors](j)), j=1..n)/n)
end:
seq(a(n), n=1..30); # Alois P. Heinz, Jan 05 2022
a[n_] := a[n] = If[n == 0, 1, Sum[a[n - j]*Sum[d*
Max[1, 2^(d - 1) - 1], {d, Divisors[j]}], {j, 1, n}]/n];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={EulerT(Vec((1 -2*x +2*x^2)/((1-x)*(1-2*x)) + O(x*x^n)))} \\ Andrew Howroyd, Nov 19 2021
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(d*
max(1, 2^(d-1)-1), d=numtheory[divisors](j)), j=1..n)/n)
end:
a:= n-> b(n)-max(1, 2^(n-1)-1):
seq(a(n), n=1..35); # Alois P. Heinz, Jan 05 2022
b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j]*Sum[d*
Max[1, 2^(d-1) - 1], {d, Divisors[j]}], {j, 1, n}]/n];
a[n_] := b[n] - Max[1, 2^(n-1)-1];
Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Mar 11 2022, Alois P. Heinz *)
Comments