A210655 -id:A210655 - cp's OEIS frontend

A053530 Expansion of e.g.f.: exp(-x - x^2/2 + x*exp(x)).

1, 0, 1, 3, 7, 35, 171, 847, 5041, 32643, 223705, 1659581, 13182159, 110802133, 984241363, 9212696235, 90477239521, 929604133343, 9969157068273, 111329454692485, 1291932988047775, 15550838026589061, 193833398512358011, 2498039016973836491

Offset: 0

N. J. A. Sloane, Jan 16 2000

nonn

The number of simple labeled graphs on n nodes whose connected components are stars. - Geoffrey Critzer, Dec 10 2011

Equivalently, the number of minimal edge covers of the complete graph K_n. - Andrew Howroyd, Aug 04 2017

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.15(b).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
Eric Weisstein's World of Mathematics, Complete Graph
Eric Weisstein's World of Mathematics, Minimal Edge Cover

Cf. A000248, A210655.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(-x -x^2/2 +x*Exp(x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019

nn = 30; a = x Exp[x]; Range[0, nn]! CoefficientList[Series[Exp[a - x^2/2! - x], {x, 0, nn}], x] (* Geoffrey Critzer, Dec 10 2011 *)
CoefficientList[Series[Exp[-x - x^2/2 + x Exp[x]], {x, 0, 30}], x] Range[0, 30]! (* Eric W. Weisstein, Aug 10 2017 *)
Table[n! Sum[1/k! (Binomial[k, n-k] 2^(k-n) (-1)^k + Sum[Binomial[k, j] Sum[j^(i-j)/(i-j)! Binomial[k-j, n-i-k+j] 2^(i-j+k-n) (-1)^(k-j), {i, j, n-k+j}], {j, k}]), {k, n}], {n, 30}] (* Eric W. Weisstein, Aug 10 2017 *)

a(n):=n!*sum((binomial(k,n-k)*2^(k-n)*(-1)^k +sum(binomial(k,j) *sum(j^(i-j)/(i-j)!*binomial(k-j,n-i-k+j)*(1/2)^(n-i-k+j)*(-1)^(k-j),i,j,n-k+j),j,1,k))/k!,k,1,n); /* Vladimir Kruchinin, Sep 10 2010 */

x='x+O('x^30); Vec(serlaplace(exp(-x-1/2*x^2+x*exp(x)))) \\ Altug Alkan, Aug 10 2017

m = 30; T = taylor(exp(-x -x^2/2 +x*exp(x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019

a(n) = n!*Sum_{k=1..n} (1/k!)*( binomial(k, n-k)*2^(k-n)*(-1)^k + Sum_{j=1..k} binomial(k,j)* (Sum_{i=j..n-k+j} (j^(i-j)/(i-j)! * binomial(k-j,n-i-k+j)*(1/2)^(n-i-k+j)*(-1)^(k-j)) ) ), n>0. - Vladimir Kruchinin, Sep 10 2010
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ n^n / (exp(r^2/2 + n*r/(1+r)) * r^n * sqrt(r^2*(1+r)/n + 2+r-1/(1+r))), where r is the root of the equation r*(exp(r)*(1+r)-1-r) = n.
(a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n)/2)))/(2*LambertW(sqrt(n)/2)).
(End)

A210654 Triangle read by rows: T(n,k) (1 <= k <= n) = number of irreducible coverings by edges of the complete bipartite graph K_{n,k}.

Original entry on oeis.org

1, 1, 2, 1, 6, 15, 1, 14, 48, 184, 1, 30, 165, 680, 2945, 1, 62, 558, 2664, 13080, 63756, 1, 126, 1827, 11032, 59605, 320292, 1748803, 1, 254, 5820, 46904, 281440, 1663248, 9791824, 58746304, 1, 510, 18177, 200232, 1379745, 8906544, 56499849, 361679040, 2361347073

Offset: 1

N. J. A. Sloane, Mar 27 2012

			Triangle begins:
  1;
  1,   2;
  1,   6,   15;
  1,  14,   48,   184;
  1,  30,  165,   680,   2945;
  1,  62,  558,  2664,  13080,   63756;
  1, 126, 1827, 11032,  59605,  320292, 1748803;
  1, 254, 5820, 46904, 281440, 1663248, 9791824, 58746304;
  ...

Alois P. Heinz, Rows n = 1..120, flattened
Ioan Tomescu, Some properties of irreducible coverings by cliques of complete multipartite graphs, J. Combin. Theory Ser. B 28 (1980), no. 2, 127--141. MR0572469 (81i:05106).

Cf. A210655.

T:= proc(p, q) option remember; `if`(p=1 or q=1, 1,
         add(binomial(q, r)   *T(p-1, q-r), r=2..q-1)
      +q*add(binomial(p-1, s) *T(p-s-1, q-1), s=0..p-2))
    end:
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Feb 10 2013

T[p_, q_] := T[p, q] = If[p == 1 || q == 1, 1, Sum[Binomial[q, r]*T[p-1, q-r], {r, 2, q-1}] + q*Sum[Binomial[p-1, s]*T[p-s-1, q-1], {s, 0, p-2}]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Mar 19 2014, after Alois P. Heinz *)

all(m) = {
mat = matrix(m, m);
for (i=1, m, for (j=1, m,
   if ((i == 1) || (j == 1), mat[i, j] = 1,
    if (i == j, mat[i, j] = i*mat[i-1,i-1] + sum(s=2,i-1, (s+1)*binomial(i,s)*mat[i-1,i-s]),
     mat[i, j] = sum(r=2, j-1, binomial(j,r)*mat[i-1,j-r]) + j*sum(s=0,i-2,binomial(i-1,s)*mat[i-s-1,j-1]));
   );
  );
);
for (i=1, m, for (j=1, i, print1(mat[i,j], ", ");); print(""););
print("");
for (i=1, m,print1(mat[i,i], ", "); );
} \\ Michel Marcus, Feb 10 2013

E.g.f.: exp(x*exp(y)+y*exp(x)-x-y-x*y)-1. - Alois P. Heinz, Feb 10 2013

A290755 Number of minimal edge covers in the n-crown graph.

Original entry on oeis.org

1, 0, 1, 5, 49, 759, 16081, 435833, 14517441, 579937319, 27203499361, 1474723875789, 91200920752129, 6365087902895747, 496792437580978449, 43025414912824996889, 4106965602739453756801, 429531453143336097416367, 48964278165034713331278529, 6055596695306076807138311717, 809134978410285605488807023681

Offset: 0

Eric W. Weisstein, Aug 09 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..60
Eric Weisstein's World of Mathematics, Crown Graph.
Eric Weisstein's World of Mathematics, Minimal Edge Cover.

Cf. A210655.

\\ by inclusion-exclusion.
B(n,k)={ my(xe=exp(x+O(x*x^n)), ye=exp(y+O(y*y^n))); n!^2*polcoef(polcoef((xe+ye-1)^k*exp(x*ye + y*xe - (x+y+x*y)),n),n)}
a(n) = {sum(k=0, n, binomial(n,k)*B(n-k,k)*(-1)^k)} \\ Andrew Howroyd, May 29 2025

a(6)-a(8) from Giovanni Resta, Aug 10 2017
a(1)-a(2) inserted by Eric W. Weisstein, Feb 14 2022
a(0)=1 prepended and a(9) onwards from Andrew Howroyd, May 29 2025

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A053530 Expansion of e.g.f.: exp(-x - x^2/2 + x*exp(x)).

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A210654 Triangle read by rows: T(n,k) (1 <= k <= n) = number of irreducible coverings by edges of the complete bipartite graph K_{n,k}.

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A290755 Number of minimal edge covers in the n-crown graph.

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