A182161 -id:A182161 - cp's OEIS frontend

A001192 Number of full sets of size n.

1, 1, 1, 2, 9, 88, 1802, 75598, 6421599, 1097780312, 376516036188, 258683018091900, 355735062429124915, 978786413996934006272, 5387230452634185460127166, 59308424712939278997978128490, 1305926814154452720947815884466579

Offset: 0

N. J. A. Sloane

A set x is full if every element of x is also a subset of x.
Equals the subpartitions of Eulerian numbers (A000295(n)=2^n-n-1); see A115728 for the definition of subpartitions of a partition. - Paul D. Hanna, Jul 03 2006
Also number of transitive rooted identity trees with n branches. - Gus Wiseman, Dec 21 2016

			Examples of full sets are 0 := {}, 1 := {0}, 2 := {1,0}, 3a := {2,1,0}, 3b := { {1}, 1, 0}, 4a := { 3a, 2, 1, 0 }.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 123, Problem 20.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..50
Bojan Bašić, Paul Ellis, Dana C. Ernst, Danijela Popović, and Nándor Sieben, Categories of impartial rulegraphs and gamegraphs, arXiv:2312.00650 [math.CO], 2023. See p. 20.
Alberto Casagrande, Carla Piazza, and Alberto Policriti, Is hyper-extensionality preservable under deletions of graph elements?, Elec. Notes Theor. Comp. Sci. (2016) Vol. 322, 103-118.
Richard Peddicord, The number of full sets with n elements, Proc. Amer. Math. Soc., 13 (1962), 825-828.
John Riordan, Letter to N. J. A. Sloane, Jul. 1970
Alexandru Ioan Tomescu, Sets as Graphs, Ph. D. Thesis, Università degli Studi di Udine, Dipartimento di Matematica e Informatica, Dottorato di Ricerca in Informatica, Dec. 2011.
Stephan Wagner, Asymptotic enumeration of extensional acyclic digraphs, in Proceedings of the SIAM Meeting on Analytic Algorithmics and Combinatorics (ANALCO12).
Gus Wiseman, Transitive rooted identity trees with n=5 branches

Cf. A000295, A004111, A115728, A182161, A279861, A279863.

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: seq(A001192(n), n=0..16); # Nathaniel Johnston, Apr 18 2012

max = 16; f[x_] := Sum[a[n]*(x^n/(1+x)^2^n), {n, 0, max}] - 1; cc = CoefficientList[ Series[f[x], {x, 0, max}], x]; Table[a[n], {n, 0, max}] /. First[ Solve[ Thread[cc == 0]]] (* Jean-François Alcover, Nov 02 2011, after Vladeta Jovovic *)

{a(n)=polcoeff(x^n-sum(k=0, n-1, a(k)*x^k*(1-x+x*O(x^n))^(2^k-k-1) ), n)} \\ Paul D. Hanna, Jul 03 2006

1 = Sum_{n>=0} a(n)*x^n/(1+x)^(2^n). E.g., 1 = 1/(1+x) + 1*x/(1+x)^2 + 1*x^2/(1+x)^4 + 2*x^3/(1+x)^8 + 9*x^4/(1+x)^16 + 88*x^5/(1+x)^32 + 1802*x^6/(1+x)^64 + ... . - Vladeta Jovovic, May 26 2005
Equivalently, a(n) = (-1)^n*C(2^n+n-1, n) - Sum_{k=0..n-1} a(k)*(-1)^(n-k)*C(2^n+2^k+n-k-1, n-k). - Paul D. Hanna, May 26 2005
G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^(2^n-n-1) = 1*(1-x)^0 + 1*x*(1-x)^0 + 1*x^2*(1-x)^1 + 2*x^3*(1-x)^4 + 9*x^3*(1-x)^11 + ... + a(n)*x^n*(1-x)^(2^n-n-1) + ... . - Paul D. Hanna, Jul 03 2006

More terms from Ryan Propper, Jun 13 2005

A182162 Triangle read by rows: number of extensional acyclic digraphs on n labeled nodes with k sources.

Original entry on oeis.org

1, 2, 12, 192, 24, 8160, 2400, 898560, 384480, 14400, 245145600, 126040320, 9777600, 50400, 159035627520, 90043269120, 9660672000, 179222400, 80640, 237882053283840, 141969202744320, 17961178152960, 547498828800, 2586608640, 802369403419852800

Offset: 1

N. J. A. Sloane, Apr 15 2012

			Triangle begins:
          1;
          2;
         12;
        192,        24;
       8160,      2400;
     898560,    384480,   14400;
  245145600, 126040320, 9777600, 50400;
  ...

S. Wagner, Asymptotic enumeration of extensional acyclic digraphs, in Proceedings of the SIAM Meeting on Analytic Algorithmics and Combinatorics (ANALCO12).

Row sums give A182161. First column is A182163. Row lengths are A182220.

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): if(vl = 0)then return NULL: fi: return vl: end: for n from 1 to 10 do seq(A182162(n,l), l=1..n); od; # Nathaniel Johnston, Apr 18 2012

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}]];
A182162[n_, l_] := Module[{vl}, vl = Sum[(-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}]; If[vl == 0, Nothing, vl]];
Table[A182162[n, l], {n, 1, 10}, {l, 1, n}] // Flatten (* Jean-François Alcover, Mar 09 2023, after Nathaniel Johnston *)

a(15)-a(25) from Nathaniel Johnston, Apr 18 2012

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

N. J. A. Sloane, Apr 15 2012

nonn

S. Wagner, Asymptotic enumeration of extensional acyclic digraphs, in Proceedings of the SIAM Meeting on Analytic Algorithmics and Combinatorics (ANALCO12).

Cf. A182161, A182162, A001192.

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

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 *)

a(8)-a(15) and removal of a(0) from Nathaniel Johnston, Apr 18 2012

cp's OEIS Frontend

A001192 Number of full sets of size n.

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A182162 Triangle read by rows: number of extensional acyclic digraphs on n labeled nodes with k sources.

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A182163 First column of A182162.

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