A101388 -id:A101388 - cp's OEIS frontend

A100548 Number of n-node labeled digraphs without endpoints.

1, 1, 1, 28, 2539, 847126, 987474781, 4267529230672, 71328353711113801, 4706871807383903992060, 1236666872833000506726110479, 1297665884376581511952494336126664, 5444003907104081585974782986977125743035, 91341304409373044577470623665964376840167100920

Offset: 0

Goran Kilibarda, Zoran Maksimovic, Vladeta Jovovic, Jan 02 2005

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A059167, A101388 (labeled case).

m:=30;
f:= func< x | Exp(3*x^2/2)*(&+[ 2^(n*(n-1))*(x*Exp(-3*x))^n/Factorial(n) : n in [0..m+2]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Mar 27 2023

m = 11;
egf = Exp[3x^2/2]*Sum[2^(n(n-1))*(x/Exp[3 x])^n/n!, {n, 0, m}];
a[n_] := SeriesCoefficient[egf, {x, 0, n}]*n!;
Table[a[n], {n, 0, m}] (* Jean-François Alcover, Feb 23 2019 *)

seq(n)={my(g=x/exp(3*x + O(x*x^n))); Vec(serlaplace(exp(3*x^2/2 + O(x*x^n))*sum(k=0, n, 2^(k*(k-1))*g^k/k!)))} \\ Andrew Howroyd, Jan 08 2020

m = 30
def f(x): return exp(3*x^2/2)*sum( 2^(n*(n-1))*(x*exp(-3*x))^n/factorial(n) for n in range(m+2) )
def A100548_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).egf_to_ogf().list()
A100548_list(m) # G. C. Greubel, Mar 27 2023

E.g.f.: exp(3/2*x^2)*(Sum_{n>=0} 2^(n*(n-1))*(x/exp(3*x))^n/n!).

Terms a(12) and beyond from Andrew Howroyd, Jan 08 2020

A101389 Number of n-vertex unlabeled oriented graphs without endpoints.

Original entry on oeis.org

1, 1, 1, 3, 21, 369, 16929, 1913682, 546626268, 406959998851, 808598348346150, 4358157210587930509, 64443771774627635711718, 2636248889492487709302815665, 300297332862557660078111708007894, 95764277032243987785712142452776403618, 85885545190811866954428990373255822969983915

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jan 14 2005

nonn

			a(3) = 3 because there are 2 distinct orientations of the triangle K_3 plus the empty graph on 3 vertices.

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A100569 (labeled case), A100548, A101388, A004110, A004108, A059166, A059167, A001174.

\\ See links in A339645 for combinatorial species functions.
oedges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, (v[i]-1)\2)}
ographsCycleIndex(n)={my(s=0); forpart(p=n, s+=permcount(p) * 3^oedges(p) * sMonomial(p)); s/n!}
ographs(n)={sum(k=0, n, ographsCycleIndex(k)*x^k) + O(x*x^n)}
trees(n,k)={sRevert(x*sv(1)/sExp(k*x*sv(1) + O(x^n)))}
cycleIndexSeries(n)={my(g=ographs(n), tr=trees(n,2), tu=tr-tr^2); sSolve( g/sExp(tu), tr )*symGroupSeries(n)}
NumUnlabeledObjsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 27 2020

\\ faster stand-alone version
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, (v[i]-1)\2)}
seq(n)={Vec(sum(k=0, n, my(s=0); forpart(p=k, s+=permcount(p) * 3^edges(p) * prod(i=1, #p, my(d=p[i]); (1-x^d)^2 + O(x*x^(n-k))) ); x^k*s/k!)/(1-x^2))} \\ Andrew Howroyd, Jan 22 2021

a(0)=1 prepended and terms a(9) and beyond from Andrew Howroyd, Dec 27 2020

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A100548 Number of n-node labeled digraphs without endpoints.

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A101389 Number of n-vertex unlabeled oriented graphs without endpoints.

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