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.

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A182161 Number of extensional acyclic digraphs on n labeled nodes.

Original entry on oeis.org

1, 1, 2, 12, 216, 10560, 1297440, 381013920, 258918871680, 398362519618560, 1366301392119014400, 10325798296570753920000, 170397664079650720884864000, 6094923358716319193283074457600, 469649545161250827117772066578739200, 77556106803568453086056722450983544320000
Offset: 0

Views

Author

N. J. A. Sloane, Apr 15 2012

Keywords

Links

Crossrefs

Cf. A001192, A003024. Row sums of A182162.

Formula

a(n) = n!*A001192(n).

A182220 Largest number k such that there exists an extensional acyclic digraph on n labeled nodes with k sources.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 58, 59, 60, 61
Offset: 1

Views

Author

Nathaniel Johnston, Apr 19 2012

Keywords

Comments

Also the length of row n of A182162.
This seems to be simply the natural numbers, with the terms in A000325 repeated.
It appears a(n+1) is the number of distinct possible heights of binary trees with n nodes. The minimum height of an n node binary tree is A000523(n), the maximum height is n-1 and all intermediate heights are possible. This conjecture is therefore equivalent to the conjectured formulas. - Yuchun Ji, Mar 22 2021
Conjecture: Partial sums of A347523, thus a(n) is the number of nonpowers of 2 <= n-1, or with offset 0: a(n) is the number of nonpowers of 2 <= n. - Omar E. Pol, Sep 30 2021

Links

Crossrefs

Cf. A083058, A182162.

Programs

  • Maple
    A001192 := proc(n) option remember: if(n=0)then return 1: fi: return add((-1)^(n-k-1)*binomial(2^k-k, n-k)*procname(k), k=0..n-1); end: A182162 := proc(n, l) local vl: vl := add((-1)^(k-l)*binomial(n, k)*binomial(k, l)*binomial(2^(n-k)-n+k, k)*k!*(n-k)!*A001192(n-k), k=l..n): return vl: end: A182220 := proc(n) local l: for l from n to 1 by -1 do if(A182162(n, l)>0)then break:fi:od: return l: end: seq(A182220(n),n=1..60);

Formula

Conjecture, for all n >= 3: a(n) = A083058(n-1) + 1 = n - 1 - A000523(n-1) = n - 1 - floor(log(2,n)). - Antti Karttunen, Aug 17 2013
Conjecture: a(1) = 0, a(n) = n - 1 - Sum_{i=1..n} sign(floor((n-1)/ 2^i)), n > 1. - Wesley Ivan Hurt, Feb 02 2014
Conjecture: a(n) = n - Sum_{k=0..n-2} A036987(k). - Paul Barry, Mar 07 2017

A279863 Number of maximal transitive finitary sets with n brackets.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 2, 2, 1, 1, 4, 3, 4, 2, 5, 6, 10, 8, 11, 11, 20, 22, 29, 36, 45, 53, 77, 83, 108, 141, 172, 208, 274, 323
Offset: 1

Views

Author

Gus Wiseman, Dec 21 2016

Keywords

Comments

A finitary set is transitive if every element is also a subset. A set system is maximal if the union is also a member.

Examples

			The a(23)=3 maximal transitive finitary sets are:
(()(())(()(()))((())(()(())))(()(())(()(())))),
(()(())((()))(((())))(()((())))(()(())((())))),
(()(())((()))(()(()))(()((())))(()(())((())))).

Links

Crossrefs

Cf. A001192, A004111, A276625, A279065, A279614, A279861.

Programs

  • Mathematica
    maxtransfins[n_]:=If[n===1,{},Select[Union@@FixedPointList[Complement[Union@@Function[fin,Cases[Complement[Subsets[fin],fin],sub_:>With[{nov=Sort[Append[fin,sub]]},nov/;Count[Union[nov,{Union@@nov}],_List,{0,Infinity}]<=n]]]/@#,#]&,{{}}],And[Count[#,_List,{0,Infinity}]===n,MemberQ[#,Union@@#]]&]];
Table[Length[maxtransfins[n]],{n,20}]

A182163 First column of A182162.

Original entry on oeis.org

1, 2, 12, 192, 8160, 898560, 245145600, 159035627520, 237882053283840, 802369403419852800, 6005354444640501350400, 98553538944200922572390400, 3514155297016560613680059596800, 270315783633381492859539110078054400, 44596108353446508026919663976179916800000
Offset: 1

Views

Author

N. J. A. Sloane, Apr 15 2012

Keywords

Links

Crossrefs

Cf. A182161, A182162, A001192.

Programs

  • Maple
    A001192 := proc(n) option remember: if(n=0)then return 1: fi: return add((-1)^(n-k-1)*binomial(2^k-k,n-k)*procname(k), k=0..n-1); end: A182163 := proc(n) return add((-1)^(k-1)*k*binomial(n,k)*binomial(2^(n-k)-n+k,k)*k!*(n-k)!*A001192(n-k), k=1..n): end: seq(A182163(n), n=1..16); # Nathaniel Johnston, Apr 18 2012
  • Mathematica
    A001192[n_] := A001192[n] = If[n == 0, 1, Sum[(-1)^(n - k - 1)*Binomial[2^k - k, n - k]*A001192[k], {k, 0, n - 1}]];
a[n_] := Sum[(-1)^(k - 1)*Binomial[n, k]*k*Binomial[2^(n - k) - n + k, k]*k!*(n - k)!*A001192[n - k], {k, 1, n}];
Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Apr 12 2023, after Nathaniel Johnston *)

Extensions

a(8)-a(15) and removal of a(0) from Nathaniel Johnston, Apr 18 2012
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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