A058864 -id:A058864 - cp's OEIS frontend

A058863 Number of connected labeled chordal graphs on n nodes with no induced path P_4; also the number of labeled trees with each vertex replaced by a clique.

1, 1, 4, 23, 181, 1812, 22037, 315569, 5201602, 97009833, 2019669961, 46432870222, 1168383075471, 31939474693297, 942565598033196, 29866348653695203, 1011335905644178273, 36446897413531401020, 1392821757824071815641, 56259101478392975833333

Offset: 1

Robert Castelo, Jan 06 2001

nonn

A subclass of chordal-comparability graphs.

Jon E. Schoenfield, Table of n, a(n) for n = 1..100
R. Castelo and N. C. Wormald, Enumeration of P4-free chordal graphs
R. Castelo and N. C. Wormald, Enumeration of P4-Free chordal graphs, Graphs and Combinatorics, 19:467-474, 2003.
M. C. Golumbic, Trivially perfect graphs, Discr. Math. 24(1) (1978), 105-107.
Venkatesan Guruswami, Enumerative aspects of certain subclasses of perfect graphs, Discrete Math. 205 (1999), 97-117.
T. H. Ma and J. P. Spinrad, Cycle-free partial orders and chordal comparability graphs, Order, 1991, 8:49-61.
E. S. Wolk, A note on the comparability graph of a tree, Proc. Am. Math. Soc., 1965, 16:17-20.

Cf. A007134, A058864, A058865.

Cf. A048802.

S:= series(-LambertW(exp(-x)-1), x, 101):
seq(coeff(S,x,j)*j!, j=1..100); # Robert Israel, Nov 30 2015

a[n_] := Sum[(-1)^(n-k)*StirlingS2[n, k]*k^(k-1), {k, 1, n}];
Array[a, 20] (* Jean-François Alcover, Dec 17 2017, after Vladeta Jovovic *)

geta(n, va, vA) = {local(k); if (n==1, return(1)); if (n==2, return(1)); return(1 + sum(k=1, n-2, binomial(n,k)*(vA[n-k] - va[n-k])));}
getA(n, va, vA) = {local(k); if (n==1, return(1)); if (n==2, return(2)); return ((va[n] + sum(k=1, n-1, k*va[k]*binomial(n,k)*vA[n-k])/n));}
both(n) = {va = vector(n); vA = vector(n); for (i=1, n, va[i] = geta(i, va, vA); vA[i] = getA(i, va, vA);); print("va_A058863=", va); print("vA_A058864=", vA);}
\\ Michel Marcus, Apr 03 2013

A058863 and A058864 satisfy:
  1) c(n) = 1 + Sum_{k=1..n-2} binomial(n, k)*(t(n-k) - c(n-k))
  2) t(n) = c(n) + Sum_{k=1..n-1} k*c(k)*binomial(n, k)*t(n-k)/n
  where c(n) (A058863) is the number of connected graphs of this type and t(n) (A058864) is the total number of such graphs.
a(n) is asymptotic to sqrt(r*(e-1))/n*(n/(e*r))^n where r = 1 - log(e-1).
E.g.f.: -LambertW(exp(-x)-1). - Vladeta Jovovic, Nov 22 2002
a(n) = Sum_{k=0..n} Stirling2(n, k)*A060356(k). Also a(n) = Sum_{k=1..n} (-1)^(n-k)*Stirling2(n, k)*k^(k-1). - Vladeta Jovovic, Sep 17 2003

Formulae edited and completed by Michel Marcus, Apr 07 2013

A349585 E.g.f. satisfies: A(x) * log(A(x)) = 1 - exp(-x).

Original entry on oeis.org

1, 1, -2, 8, -59, 642, -9112, 158839, -3279880, 78250188, -2117569181, 64082989720, -2144319848772, 78609355884893, -3133061858717806, 134884905211588892, -6238095343894356675, 308427209934965151158, -16234730389499986865092, 906409067599064528054343

Offset: 0

Seiichi Manyama, Nov 22 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..378
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A120980, A349561, A349583.

Cf. A008277, A058864, A349527, A349528.

b:= proc(n, m) option remember; `if`(n=0,
     (m-1)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> (-1)^(n-1)*b(n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 03 2022

a[n_] := (-1)^(n - 1) * Sum[If[k == 1, 1, (k - 1)^(k - 1)]*StirlingS2[n, k], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Nov 23 2021 *)

a(n) = (-1)^(n-1)*sum(k=0, n, (k-1)^(k-1)*stirling(n, k, 2));

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(1-exp(-x)))))

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-k+1)^(k-1)*x^k/prod(j=1, k, 1+j*x)))

a(n) = (-1)^(n-1) * Sum_{k=0..n} (k-1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = exp( LambertW(1 - exp(-x)) ).
G.f.: Sum_{k>=0} (-k+1)^(k-1) * x^k/Product_{j=1..k} (1 + j*x).
a(n) ~ -(-1)^n * sqrt(1 + exp(1)) * n^(n-1) / (exp(n+1) * (log(1 + exp(1)) - 1)^(n - 1/2)). - Vaclav Kotesovec, Dec 05 2021

A196555 O.g.f.: Sum_{n>=0} 2(n+2)^(n-1) x^n / Product_{k=1..n} (1+k*x).

Original entry on oeis.org

1, 2, 6, 28, 186, 1614, 17332, 222254, 3317326, 56532264, 1083571422, 23081180918, 541047188936, 13843339479298, 383952455939662, 11475711580482268, 367729128426998450, 12577206203908139494, 457341567152354085700, 17619050162270848917366

Offset: 0

Paul D. Hanna, Oct 03 2011

nonn

			O.g.f.: A(x) = 1 + 2*x + 6*x^2 + 28*x^3 + 186*x^4 + 1614*x^5 +...
where the o.g.f. is given by:
A(x) = 1 + 2*3^0*x/(1+x) + 2*4^1*x^2/((1+x)*(1+2*x)) + 2*5^2*x^3/((1+x)*(1+2*x)*(1+3*x)) + 2*6^3*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) +...
E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 28*x^3/3! + 186*x^4/4! + 1614*x^5/5! +...
where the e.g.f. is given by:
A(x)^(1/2) = 1 + x + 2*x^2/2! + 8*x^3/3! + 49*x^4/4! + 402*x^5/5! + 4144*x^6/6! +...+ A058864(n)*x^n/n! +...

Seiichi Manyama, Table of n, a(n) for n = 0..399

Cf. A058864, A196556, A196557.

CoefficientList[Series[E^(-2*LambertW[E^(-x)-1]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)

{a(n)=polcoeff(sum(m=0, n, 2*(m+2)^(m-1)*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}

{A058864(n)=polcoeff(sum(m=0, n, (m+1)^(m-1)*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}
{a(n) = sum(k=0,n,binomial(n,k)*A058864(n-k)*A058864(k))}

a(n)=sum(k=0, n, (-1)^(n-k)*stirling(n, k, 2)*2*(k+2)^(k-1));

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2*lambertw(exp(-x)-1)))) \\ Seiichi Manyama, Nov 21 2021

E.g.f.: exp(-2*LambertW(exp(-x)-1)).
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*2*(k+2)^(k-1).
a(n) = Sum_{k=0..n} C(n,k)*A058864(n-k)*A058864(k); exponential convolution of A058864, which is the number of labeled chordal graphs (connected or not) on n nodes with no induced path P_4.
a(n) ~ 2*sqrt(exp(1)-1) * n^(n-1) / (exp(n-2) * (1-log(exp(1)-1))^(n-1/2)). - Vaclav Kotesovec, Jul 09 2013

A196556 O.g.f.: Sum_{n>=0} 3(n+3)^(n-1) x^n / Product_{k=1..n} (1+k*x).

Original entry on oeis.org

1, 3, 12, 66, 483, 4476, 50454, 671649, 10328118, 180341094, 3527385345, 76435691250, 1818255212490, 47118807865863, 1321527658352016, 39889359465259446, 1289471521115731611, 44450463108654209136, 1627806562174453037802

Offset: 0

Paul D. Hanna, Oct 03 2011

nonn

			O.g.f.: A(x) = 1 + 3*x + 12*x^2 + 66*x^3 + 483*x^4 + 4476*x^5 +...
where the o.g.f. is given by:
A(x) = 1 + 3*4^0*x/(1+x) + 3*5^1*x^2/((1+x)*(1+2*x)) + 3*6^2*x^3/((1+x)*(1+2*x)*(1+3*x)) + 3*7^3*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) +...
E.g.f.: A(x) = 1 + 3*x + 12*x^2/2! + 66*x^3/3! + 483*x^4/4! + 4476*x^5/5! +...
where the e.g.f. is given by:
A(x)^(1/3) = 1 + x + 2*x^2/2! + 8*x^3/3! + 49*x^4/4! + 402*x^5/5! + 4144*x^6/6! +...+ A058864(n)*x^n/n! +...

Seiichi Manyama, Table of n, a(n) for n = 0..399

Cf. A058864, A196555, A196557.

CoefficientList[Series[E^(-3*LambertW[E^(-x)-1]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)

{a(n)=polcoeff(sum(m=0, n, 3*(m+3)^(m-1)*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}

/* E.g.f. = G(x)^3 where G(x) = e.g.f. of A058864 */
{A058864(n)=polcoeff(sum(m=0, n, (m+1)^(m-1)*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}
{a(n)=n!*polcoeff(sum(k=0,n,A058864(k)*x^k/k!+x*O(x^n))^3,n)}

a(n)=sum(k=0, n, (-1)^(n-k)*stirling(n, k, 2)*3*(k+3)^(k-1));

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-3*lambertw(exp(-x)-1)))) \\ Seiichi Manyama, Nov 21 2021

E.g.f.: exp(-3*LambertW(exp(-x)-1)).
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*3*(k+3)^(k-1).
E.g.f.: A(x) = G(x)^3 where G(x) = e.g.f. of A058864, which is the number of labeled chordal graphs (connected or not) on n nodes with no induced path P_4.
a(n) ~ 3*sqrt(exp(1)-1) * n^(n-1) / (exp(n-3) * (1-log(exp(1)-1))^(n-1/2)). - Vaclav Kotesovec, Jul 09 2013

A058865 Irregular table a(n,k) = number of connected labeled chordal graphs on n nodes with k edges, containing no induced path P_4, for n >= 1, 1 <= k <= n*(n-1)/2, read by rows; also the number of labeled trees with each vertex replaced by a clique.

Original entry on oeis.org

1, 0, 3, 1, 0, 0, 4, 12, 6, 1, 0, 0, 0, 5, 30, 75, 30, 30, 10, 1, 0, 0, 0, 0, 6, 60, 270, 360, 435, 270, 255, 80, 60, 15, 1, 0, 0, 0, 0, 0, 7, 105, 735, 1925, 2940, 3591, 4165, 2310, 2520, 1925, 882, 630, 175, 105, 21, 1, 0, 0, 0, 0, 0, 0, 8, 168, 1680, 7280, 16800

Offset: 1

Robert Castelo, Jan 06 2001

A subclass of chordal-comparability graphs.

			The table starts:
   n | a(n, 1 <= k <= n(n-1)/2)
  ---+---------------------------
   1 | ()    (row length = 0: empty row)
   2 | 1
   3 | 0, 3, 1
   4 | 0, 0, 4, 12, 6, 1
   5 | 0, 0, 0, 5, 30, 75, 30, 30, 10, 1
  ...

G. C. Greubel, Rows n = 2..30 of the irregular triangle, flattened
R. Castelo and N. C. Wormald, Enumeration of P4-free chordal graphs.
R. Castelo and N. C. Wormald, Enumeration of P4-Free chordal graphs, Graphs and Combinatorics, 19:467-474, 2003.
M. C. Golumbic, Trivially perfect graphs, Discr. Math. 24(1) (1978), 105-107.
T. H. Ma and J. P. Spinrad, Cycle-free partial orders and chordal comparability graphs, Order, 1991, 8:49-61.
E. S. Wolk, A note on the comparability graph of a tree, Proc. Am. Math. Soc., 1965, 16:17-20.

Cf. A000217 (row lengths, up to offset), A000292, A007134, A058863, A058864.

Cf. A356916.

T[n_, k_]:= T[n, k]= If[k==Binomial[n, 2], 1, Sum[Binomial[n, j]*(A[n-j, k-j*(2*n -1-j)/2] - T[n-j, k-j*(2*n-1-j)/2]), {j, n-2}]]; (* T = A058865 *)
A[n_, k_]:= A[n, k]= T[n, k] + Sum[Sum[Binomial[n-1, j-1]*T[j, m]*A[n-j, k-m], {j, n-1}], {m, 0, k}]; (* A = A356916 *)
Table[T[n, k], {n,2,12}, {k,Binomial[n, 2]}]//Flatten (* G. C. Greubel, Sep 03 2022 *)

A58865=Map(); A058865(n,q) = if( q < n-1 || q >= n*(n-1)\2, q==n*(n-1)\2, mapisdefined(A58865, [n,q], &q), q, mapput(A58865, [n,q], q = sum(k=1,n-2, binomial(n,k)*(A356916(n-k, q - k*(k-1)/2 - k*(n-k)) - A058865(n-k, q - k*(k-1)\2 - (n-k)*k)))); q) \\ A356916 "outsourced" by M. F. Hasler, Sep 26 2022
[[A058865(n,k)| k<-[1..n*(n-1)/2]] | n<-[1..7]] \\ M. F. Hasler, Sep 03 2022

@CachedFunction
def T(n,k): # T = A058865
    if (k==binomial(n,2)): return 1
    else: return sum( binomial(n,j)*( A(n-j, k-j*(2*n-1-j)/2) - T(n-j, k-j*(2*n-1-j)/2) ) for j in (1..n-2) )
@CachedFunction
def A(n,k): # A = A356916
    return T(n,k) + sum(sum( binomial(n-1,j-1)*T(j,m)*A(n-j,k-m) for j in (1..n-1) ) for m in (0..k) )
flatten([[T(n,k) for k in (1..binomial(n,2))] for n in (2..12)]) # G. C. Greubel, Sep 03 2022

Let A(n,k) be the total number of labeled P_4 - free chordal graphs on n vertices and q edges (= A356916), then:
a(n,q) = Sum_{k=1..n-2} binomial(n,k)*(A(n-k, q - k(k-1)/2 - k(n-k)) - a(n-k, q - k(k-1)/2 - k(n-k))) for q < n(n-1)/2 =: T(n), a(n, T(n)) = 1. [Corrected by M. F. Hasler, Sep 03 2022]
A(n,q) = a(n,q) + Sum_{k = 1..n-1} binomial(n-1, k-1)*Sum_{l = k-1..min(k(k-1)/2, q)} a(k,l)*A(n-k,q-l). [Simplified by M. F. Hasler, Sep 03 2022]
Particular values: a(n,k) = 0 for q < n-1; a(n, T(n)) = 1; a(n,n-1) = n; a(n, T(n)-1) = n(n-1)/2 for n > 2, a(n, n) = a(n, T(n)-2) = n(n-1)(n-2)/2 for n > 3. - M. F. Hasler, Sep 03 2022
From G. C. Greubel, Sep 03 2022: (Start)
a(n, binomial(n,2) - 1) = A000217(n+1) - [n=2], n >= 2.
a(n, n) = 3*A000292(n) - 2*[n=3], n >= 3.
Sum_{k=1..binomial(n,2)} a(n, k) = A058863(n). (End)

Typo in a(6,11) corrected by G. C. Greubel, Sep 03 2022

A196557 O.g.f.: Sum_{n>=0} 4(n+4)^(n-1) x^n / Product_{k=1..n} (1+k*x).

Original entry on oeis.org

1, 4, 20, 128, 1036, 10308, 122560, 1701092, 27053556, 485683128, 9723771156, 214934627476, 5201286731560, 136818097071820, 3888121468512308, 118737900886653664, 3878569457507036988, 134960059001226137588, 4984357865462772982112

Offset: 0

Paul D. Hanna, Oct 03 2011

nonn

			O.g.f.: A(x) = 1 + 4*x + 20*x^2 + 128*x^3 + 1036*x^4 + 10308*x^5 +...
where the o.g.f. is given by:
A(x) = 1 + 4*5^0*x/(1+x) + 4*6^1*x^2/((1+x)*(1+2*x)) + 4*7^2*x^3/((1+x)*(1+2*x)*(1+3*x)) + 4*8^3*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) +...
E.g.f.: A(x) = 1 + 4*x + 20*x^2/2! + 128*x^3/3! + 1036*x^4/4! + 10308*x^5/5! +...
where the e.g.f. is given by:
A(x)^(1/2) = 1 + 2*x + 6*x^2/2! + 28*x^3/3! + 186*x^4/4! + 1614*x^5/5! + 17332*x^6/6! +...+ A196555(n)*x^n/n! +...
A(x)^(1/4) = 1 + x + 2*x^2/2! + 8*x^3/3! + 49*x^4/4! + 402*x^5/5! + 4144*x^6/6! +...+ A058864(n)*x^n/n! +...

Seiichi Manyama, Table of n, a(n) for n = 0..399

Cf. A058864, A196555, A196556.

CoefficientList[Series[E^(-4*LambertW[E^(-x)-1]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)

{a(n)=polcoeff(sum(m=0, n, 4*(m+4)^(m-1)*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}

/* E.g.f. = G(x)^4 where G(x) = e.g.f. of A058864 */
{A058864(n)=polcoeff(sum(m=0, n, (m+1)^(m-1)*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}
{a(n)=n!*polcoeff(sum(k=0,n,A058864(k)*x^k/k!+x*O(x^n))^4,n)}

a(n)=sum(k=0, n, (-1)^(n-k)*stirling(n, k, 2)*4*(k+4)^(k-1));

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-4*lambertw(exp(-x)-1)))) \\ Seiichi Manyama, Nov 21 2021

E.g.f.: exp(-4*LambertW(exp(-x)-1)).
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*4*(k+4)^(k-1).
E.g.f.: A(x) = G(x)^4 where G(x) = e.g.f. of A058864, which is the number of labeled chordal graphs (connected or not) on n nodes with no induced path P_4.
a(n) ~ 4*sqrt(exp(1)-1)*n^(n-1)/(exp(n-4)*(1-log(exp(1)-1))^(n-1/2)). - Vaclav Kotesovec, Jul 09 2013

A349527 a(n) = Sum_{k=0..n} (-1)^(n-k) * (2k+1)^(k-1) Stirling2(n,k).

Original entry on oeis.org

1, 1, 4, 35, 469, 8502, 194807, 5402497, 175985390, 6587650497, 278674144201, 13148017697608, 684554867667117, 38988819551585477, 2411411875573335044, 160951864352781351959, 11531509389384310870257

Offset: 0

Seiichi Manyama, Nov 20 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..356
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A008277, A058864, A196555, A349524, A349528.

a[n_] := Sum[(-1)^(n - k) * (2*k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Nov 27 2021 *)

a(n) = sum(k=0, n, (-1)^(n-k)*(2*k+1)^(k-1)*stirling(n, k, 2));

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*(exp(-x)-1))/2)))

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (2*k+1)^(k-1)*x^k/prod(j=1, k, 1+j*x)))

E.g.f. satisfies: log(A(x)) = (1 - exp(-x)) * A(x)^2.
E.g.f.: exp( -LambertW(2 * (exp(-x) - 1))/2 ).
G.f.: Sum_{k>=0} (2*k+1)^(k-1) * x^k/Product_{j=1..k} (1 + j*x).
a(n) ~ sqrt(2*exp(1) - 1) * sqrt(log(2/(2-exp(-1)))) * n^(n-1) / (2 * exp(n - 1/2) * (1 + log(2/(2*exp(1) - 1)))^n). - Vaclav Kotesovec, Nov 21 2021

A349528 a(n) = Sum_{k=0..n} (-1)^(n-k) * (3k+1)^(k-1) Stirling2(n,k).

Original entry on oeis.org

1, 1, 6, 80, 1645, 45962, 1627080, 69817575, 3522349232, 204343964292, 13403304111515, 980876342339456, 79235384391436316, 7003257362607771709, 672285536392973397658, 69656231091367157111844, 7747832754070176901631621

Offset: 0

Seiichi Manyama, Nov 20 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..336
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A008277, A058864, A196556, A349525, A349527.

a[n_] := Sum[(-1)^(n - k)*(3*k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Nov 21 2021 *)

a(n) = sum(k=0, n, (-1)^(n-k)*(3*k+1)^(k-1)*stirling(n, k, 2));

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(3*(exp(-x)-1))/3)))

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (3*k+1)^(k-1)*x^k/prod(j=1, k, 1+j*x)))

E.g.f. satisfies: log(A(x)) = (1 - exp(-x)) * A(x)^3.
E.g.f.: exp( -LambertW(3 * (exp(-x) - 1))/3 ).
G.f.: Sum_{k>=0} (3*k+1)^(k-1) * x^k/Product_{j=1..k} (1 + j*x).
a(n) ~ sqrt(3*exp(1) - 1) * sqrt(log(3/(3-exp(-1)))) * n^(n-1) / (3 * exp(n - 1/3) * (1 + log(3/(3*exp(1) - 1)))^n). - Vaclav Kotesovec, Nov 21 2021

A349558 E.g.f. satisfies: log(A(x)) = (1 - exp(-xA(x))) A(x).

Original entry on oeis.org

1, 1, 4, 32, 393, 6547, 138046, 3525853, 105832964, 3651748332, 142429413387, 6196895235709, 297571887174040, 15632879134292045, 891910713837242092, 54919409605089141532, 3630105859259972654905, 256374187841461047791587

Offset: 0

Seiichi Manyama, Nov 21 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..360

Cf. A058864, A349557.

a[n_] := Sum[(-1)^(n - k) * (n + k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 22 2021 *)

a(n) = sum(k=0, n, (-1)^(n-k)*(n+k+1)^(k-1)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} (-1)^(n-k) * (n+k+1)^(k-1) * Stirling2(n,k).

a(n) ~ sqrt((s-1)*s^3 / (1 + r*(2*s - 3)*s - r^2*(s-1)*s^2)) * n^(n-1) / (exp(n) * r^(n -1/2)), where r = 0.2202409288542107090687589144963703329896230236509... and s = 1.7315644042495989781932730410872588555151921253414... are roots of the system of equations s = s/exp(r*s) + log(s), (s-1)/s - (1 - r*s)/exp(r*s) = 0. - Vaclav Kotesovec, Nov 22 2021

A355782 E.g.f. satisfies log(A(x)) = 2 * (1 - exp(-x)) * A(x).

Original entry on oeis.org

1, 2, 10, 94, 1314, 24494, 572418, 16109678, 530772610, 20049256686, 854425665410, 40560727143534, 2122785621956226, 121440903560075246, 7539867236251002242, 504946360197545803630, 36284349255747713008770, 2784785703026225861819118

Offset: 0

Seiichi Manyama, Jul 16 2022

nonn

Cf. A058864, A351277, A355789.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*(exp(-x)-1)))))

a(n) = sum(k=0, n, (-1)^(n-k)*2^k*(k+1)^(k-1)*stirling(n, k, 2));

E.g.f.: exp( -LambertW(2 * (exp(-x) - 1)) ).
a(n) = Sum_{k=0..n} (-1)^(n-k) * 2^k * (k+1)^(k-1) * Stirling2(n,k).
From Vaclav Kotesovec, Jul 18 2022: (Start)
E.g.f.: LambertW(2 * (exp(-x) - 1)) / (2 * (exp(-x) - 1)).
a(n) ~ sqrt(2*exp(1) - 1) * sqrt(log(2/(2 - exp(-1)))) * n^(n-1) / (exp(n-1) * (log(2/(2*exp(1)-1)) + 1)^n). (End)

cp's OEIS Frontend

A058863 Number of connected labeled chordal graphs on n nodes with no induced path P_4; also the number of labeled trees with each vertex replaced by a clique.

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A349585 E.g.f. satisfies: A(x) * log(A(x)) = 1 - exp(-x).

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A196555 O.g.f.: Sum_{n>=0} 2*(n+2)^(n-1) * x^n / Product_{k=1..n} (1+k*x).

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A196556 O.g.f.: Sum_{n>=0} 3*(n+3)^(n-1) * x^n / Product_{k=1..n} (1+k*x).

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A058865 Irregular table a(n,k) = number of connected labeled chordal graphs on n nodes with k edges, containing no induced path P_4, for n >= 1, 1 <= k <= n*(n-1)/2, read by rows; also the number of labeled trees with each vertex replaced by a clique.

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A196557 O.g.f.: Sum_{n>=0} 4*(n+4)^(n-1) * x^n / Product_{k=1..n} (1+k*x).

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A196555 O.g.f.: Sum_{n>=0} 2(n+2)^(n-1) x^n / Product_{k=1..n} (1+k*x).

A196556 O.g.f.: Sum_{n>=0} 3(n+3)^(n-1) x^n / Product_{k=1..n} (1+k*x).

A196557 O.g.f.: Sum_{n>=0} 4(n+4)^(n-1) x^n / Product_{k=1..n} (1+k*x).

A349527 a(n) = Sum_{k=0..n} (-1)^(n-k) * (2k+1)^(k-1) Stirling2(n,k).

A349528 a(n) = Sum_{k=0..n} (-1)^(n-k) * (3k+1)^(k-1) Stirling2(n,k).

A349558 E.g.f. satisfies: log(A(x)) = (1 - exp(-xA(x))) A(x).