A002031 -id:A002031 - cp's OEIS frontend

A047863 Number of labeled graphs with 2-colored nodes where black nodes are only connected to white nodes and vice versa.

1, 2, 6, 26, 162, 1442, 18306, 330626, 8488962, 309465602, 16011372546, 1174870185986, 122233833963522, 18023122242478082, 3765668654914699266, 1114515608405262434306, 467221312005126294077442, 277362415313453291571118082, 233150477220213193598856331266

Offset: 0

N. J. A. Sloane

Row sums of A111636. - Peter Bala, Sep 30 2012
Column 2 of Table 2 in Read. - Peter Bala, Apr 11 2013
It appears that 5 does not divide a(n), that a(n) is even for n>0, that 3 divides a(2n) for n>0, that 7 divides a(6n+5), and that 13 divides a(12n+3). - Ralf Stephan, May 18 2013

			For n=2, {1,2 black, not connected}, {1,2 white, not connected}, {1 black, 2 white, not connected}, {1 black, 2 white, connected}, {1 white, 2 black, not connected}, {1 white, 2 black, connected}.
G.f. = 1 + 2*x + 6*x^2 + 26*x^3 + 162*x^4 + 1442*x^5 + 18306*x^6 + ...

H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 79, Eq. 3.11.2.

T. D. Noe, Table of n, a(n) for n = 0..50
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
S. R. Finch, Bipartite, k-colorable and k-colored graphs
S. R. Finch, Bipartite, k-colorable and k-colored graphs, June 5, 2003. [Cached copy, with permission of the author]
A. Gainer-Dewar and I. M. Gessel, Enumeration of bipartite graphs and bipartite blocks, arXiv:1304.0139 [math.CO], 2013.
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410-414.
R. P. Stanley, Acyclic orientation of graphs Discrete Math. 5 (1973), 171-178. North Holland Publishing Company.
Martin Svatoš, Peter Jung, Jan Tóth, Yuyi Wang, and Ondřej Kuželka, On Discovering Interesting Combinatorial Integer Sequences, arXiv:2302.04606 [cs.LO], 2023, p. 17.
Eric Weisstein's World of Mathematics, k-Colorable Graph
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 88, Eq. 3.11.2.

Column k=2 of A322280.
Cf. A135079 (variant).
Cf. A000683, A001831, A002031, A033995, A047864, A052332, A111636, A191371, A223887.

A047863:= func< n | (&+[Binomial(n,k)*2^(k*(n-k)): k in [0..n]]) >;
[A047863(n): n in [0..40]]; // G. C. Greubel, Nov 03 2024

Table[Sum[Binomial[n,k]2^(k(n-k)),{k,0,n}],{n,0,20}] (* Harvey P. Dale, May 09 2012 *)
nmax = 20; CoefficientList[Series[Sum[E^(2^k*x)*x^k/k!, {k, 0, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 05 2019 *)

{a(n)=n!*polcoeff(sum(k=0,n,exp(2^k*x +x*O(x^n))*x^k/k!),n)} \\ Paul D. Hanna, Nov 27 2007

{a(n)=polcoeff(sum(k=0, n, x^k/(1-2^k*x +x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Mar 08 2008

N=66; x='x+O('x^N); egf = sum(n=0, N, exp(2^n*x)*x^n/n!);
Vec(serlaplace(egf))  \\ Joerg Arndt, May 04 2013

from sympy import binomial
def a(n): return sum([binomial(n, k)*2**(k*(n - k)) for k in range(n + 1)]) # Indranil Ghosh, Jun 03 2017

def A047863(n): return sum(binomial(n,k)*2^(k*(n-k)) for k in range(n+1))
[A047863(n) for n in range(41)] # G. C. Greubel, Nov 03 2024

a(n) = Sum_{k=0..n} binomial(n, k)*2^(k*(n-k)).
a(n) = 4 * A000683(n) + 2. - Vladeta Jovovic, Feb 02 2000
E.g.f.: Sum_{n>=0} exp(2^n*x)*x^n/n!. - Paul D. Hanna, Nov 27 2007
O.g.f.: Sum_{n>=0} x^n/(1 - 2^n*x)^(n+1). - Paul D. Hanna, Mar 08 2008
From Peter Bala, Apr 11 2013: (Start)
Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)) = 1 + x + x^2/(2!*2) + x^3/(3!*2^3) + .... Then a generating function is E(x)^2 = 1 + 2*x + 6*x^2/(2!*2) + 26*x^3/(3!*2^3) + .... In general, E(x)^k, k = 1, 2, ..., is a generating function for labeled k-colored graphs (see Stanley). For other examples see A191371 (k = 3) and A223887 (k = 4).
If A(x) = 1 + 2*x + 6*x^2/2! + 26*x^3/3! + ... denotes the e.g.f. for this sequence then sqrt(A(x)) = 1 + x + 2*x^2/2! + 7*x^3/3! + ... is the e.g.f. for A047864, which counts labeled 2-colorable graphs. (End)
a(n) ~ c * 2^(n^2/4+n+1/2)/sqrt(Pi*n), where c = Sum_{k = -infinity..infinity} 2^(-k^2) = EllipticTheta[3, 0, 1/2] = 2.128936827211877... if n is even and c = Sum_{k = -infinity..infinity} 2^(-(k+1/2)^2) = EllipticTheta[2, 0, 1/2] = 2.12893125051302... if n is odd. - Vaclav Kotesovec, Jun 24 2013

Better description from Christian G. Bower, Dec 15 1999

A001831 Number of labeled graded partially ordered sets with n elements of height at most 1.

Original entry on oeis.org

1, 1, 3, 13, 87, 841, 11643, 227893, 6285807, 243593041, 13262556723, 1014466283293, 109128015915207, 16521353903210521, 3524056001906654763, 1059868947134489801413, 449831067019305308555487, 269568708630308018001547681, 228228540531327778410439620963

Offset: 0

N. J. A. Sloane

Labeled posets where for all a,b,c in the set, do not have a

Number of labeled digraphs with n vertices with no directed path of length 2. Number of n X n {0,1} matrices A such that A^2 = 0. - Michael Somos, Jul 28 2013

Number of relations on n labeled nodes that are simultaneously transitive and antitransitive. - Peter Kagey, Feb 14 2021

			1 + x + 3*x^2 + 13*x^3 + 87*x^4 + 841*x^5 + 11643*x^6 + 227893*x^7 + ...

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
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.
A. Motzek and R. Möller, Exploiting Innocuousness in Bayesian Networks, Preprint, Australasian Joint Conference on Artificial Intelligence AI 2015: Advances in Artificial Intelligence, pp. 411-423.
Zvi Rosen and Yan X. Zhang, Convex Neural Codes in Dimension 1, arXiv:1702.06907 [math.CO], 2017. Mentions this sequence.
Index entries for sequences related to posets

Cf. A002031, A047863, A052332, A001833, A048194.
Row sums of A052296.
Cf. variants: A135753, A135754.

A001831 := proc(n)
    add(binomial(n,k)*(2^k-1)^(n-k),k=0..n) ;
end proc:
seq(A001831(n),n=0..10) ; # R. J. Mathar, Mar 08 2021

Join[{1}, Table[Sum[Binomial[n,k](2^k-1)^(n-k),{k,n}],{n,20}]] (* Harvey P. Dale, Jan 05 2012 *)

{a(n)=n!*polcoeff(sum(k=0,n,exp((2^k-1)*x)*x^k/k!),n)} \\ Paul D. Hanna, Nov 27 2007

{a(n)=polcoeff(sum(k=0, n, x^k/(1-(2^k-1)*x +x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Sep 15 2009

a(n) = Sum((-1)^k*C(n, k)*A047863(k), k=0..n).
a(n) = Sum_{k=0..n} binomial(n, k)*(2^k-1)^(n-k). - Vladeta Jovovic, Apr 04 2003
E.g.f.: Sum_{n>=0} exp((2^n-1)*x) * x^n/n!. - Paul D. Hanna, Nov 27 2007 [correction made by Paul D. Hanna, Mar 08 2021]
O.g.f.: Sum_{n>=0} x^n/(1 - (2^n - 1)*x)^(n+1) = Sum_{n>=0} a(n)*x^n. - Paul D. Hanna, Sep 15 2009
a(n) ~ c * 2^(n^2/4 + n + 1/2) / sqrt(Pi*n), where c = JacobiTheta3(0,1/2) = EllipticTheta[3, 0, 1/2] = 2.1289368272118771586694585485449... if n is even, and c = JacobiTheta2(0,1/2) = EllipticTheta[2, 0, 1/2] = 2.1289312505130275585916134025753... if n is odd. - Vaclav Kotesovec, Mar 10 2014

More terms, formula and comments from Christian G. Bower, Dec 15 1999
Last 4 terms corrected by Vladeta Jovovic, Apr 04 2003
Comments corrected by Joel B. Lewis, Mar 28 2011

A001832 Number of labeled connected bipartite graphs on n nodes.

Original entry on oeis.org

1, 1, 3, 19, 195, 3031, 67263, 2086099, 89224635, 5254054111, 426609529863, 47982981969979, 7507894696005795, 1641072554263066471, 502596525992239961103, 216218525837808950623459, 130887167385831881114006475, 111653218763166828863141636911

Offset: 1

N. J. A. Sloane

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 406.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
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 = 1..50
F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68.
F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68. (Annotated scanned copy)
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
A. Nymeyer and R. W. Robinson, Tabulation of the Numbers of Labeled Bipartite Blocks and Related Classes of Bicolored Graphs, 1982 [Annotated scanned copy of unpublished MS and letter from R.W.R.]
Eric Weisstein's World of Mathematics, n-Colorable Graph
Eric Weisstein's World of Mathematics, n-Chromatic Graph

Row sums of A228861.
The unlabeled version is A005142.
Cf. A002031, A047863, A047864.

mx = 17; s = Sum[ Binomial[n, k] 2^(k (n - k)) x^n/n!, {n, 0, mx}, {k, 0, n}] ; Range[0, mx]! CoefficientList[ Series[ Log[s]/2, {x, 0, mx}], x] (* Geoffrey Critzer, May 10 2011 *)

seq(n)=Vec(serlaplace(log(sum(k=0, n, exp(2^k*x + O(x*x^n))*x^k/k!))/2)) \\ Andrew Howroyd, Sep 26 2018

E.g.f.: log(A(x))/2 where A(x) is e.g.f. of A047863.

a(n) = A002031(n)/2, for n > 1. - Geoffrey Critzer, May 10 2011

More terms from Vladeta Jovovic, Apr 12 2003

A052332 Number of labeled digraphs where every node has indegree 0 or outdegree 0 and no isolated nodes.

Original entry on oeis.org

1, 0, 2, 6, 50, 510, 7682, 161406, 4747010, 194342910, 11084390402, 881008805886, 97779099906050, 15178191426486270, 3302331237256396802, 1008694542117649154046, 433286992912494943469570

Offset: 0

Christian G. Bower, Dec 15 1999

nonn

Also labeled graphs with 2-colored nodes with no isolated nodes where black nodes are only connected to white nodes and vice versa.

Cf. A001831 (binomial transform), A002031, A047863.

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*A001831(k)

a(n) = Sum_{k=0..n} binomial(n, k)*(2^k-2)^(n-k). - Vladeta Jovovic, Apr 04 2003

Last 4 terms corrected by Vladeta Jovovic, Apr 04 2003

A007776 Number of connected posets with n elements of height 1.

Original entry on oeis.org

1, 2, 4, 10, 27, 88, 328, 1460, 7799, 51196, 422521, 4483460, 62330116, 1150504224, 28434624153, 945480850638, 42417674401330, 2572198227615998, 211135833162079184, 23487811567341121158, 3545543330739039981738, 727053904070651775719646

Offset: 2

Georg Wambach (gw(AT)informatik.Uni-Koeln.de)

nonn

Inverse Euler transform of A048194 and A049312. - Detlef Pauly (dettodet(AT)yahoo.de) and Vladeta Jovovic, Jul 25 2003
Essentially the same as A318870. - Georg Fischer, Oct 02 2018
Number of connected digraphs on n unlabeled nodes where every node has indegree 0 or outdegree 0 and there are no isolated nodes.  - Andrew Howroyd, Oct 03 2018

Andrew Howroyd, Table of n, a(n) for n = 2..50 (terms 2..40 from Alois P. Heinz)
N. J. A. Sloane, Transforms
J. Textor, A. Idelberger, and M. Liskiewicz, Learning from Pairwise Marginal Independencies, arXiv:1508.00280 [cs.AI], 2015.
Index entries for sequences related to posets

Cf. A005142, A002031 (labeled case), A048194, A049312, A055192, A318870, column 1 of A342500.

mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten @ Table[Map[ Function[{p}, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/ Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n - k]];
b[d_] := Sum[A[n, d - n], {n, 0, d}];
EULERi[Array[b, 30]] // Rest (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz in A049312 *)

Inverse Euler transform of A055192. - Andrew Howroyd, Oct 03 2018

More terms from Vladeta Jovovic, Jul 25 2003

Offset corrected by Andrew Howroyd, Oct 03 2018

A002032 Number of n-colored connected graphs on n labeled nodes.

Original entry on oeis.org

1, 1, 2, 24, 912, 87360, 19226880, 9405930240, 10142439229440, 24057598104207360, 125180857812868300800, 1422700916050060841779200, 35136968950395142864227532800, 1876028272361273394915958613606400, 215474119792145796020405035320528076800

Offset: 0

N. J. A. Sloane

nonn

Every connected graph on n nodes can be colored with n colors in exactly n! ways, so this sequence is just n! * A001187(n). - Andrew Howroyd, Dec 03 2018

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).

Andrew Howroyd, Table of n, a(n) for n = 0..50
R. C. Read, E. M. Wright, Colored graphs: A correction and extension, Canad. J. Math. 22 1970 594-596.

Cf. A002027. A002028, A002029, A002030, A002031.

Cf. A001187, A322278, A322279, A322280.

(* b = A001187 *) b[n_] := b[n] = If[n == 0, 1, 2^(n(n-1)/2) - Sum[k* Binomial[n, k]*2^((n-k)(n-k-1)/2)*b[k], {k, 1, n-1}]/n];
a[n_] := n! b[n];
Array[a, 14] (* Jean-François Alcover, Aug 16 2019, using Alois P. Heinz's code for A001187 *)

seq(n) = {Vec(serlaplace(serlaplace(1 + log(sum(k=0, n, 2^binomial(k, 2)*x^k/k!, O(x*x^n))))))} \\ Andrew Howroyd, Dec 03 2018

a(n) = n!*A001187(n). - Andrew Howroyd, Dec 03 2018
Define M_0(k)=1, M_n(0)=0, M_n(k) = Sum_{r=0..n} C(n,r)*2^(r*(n-r))*M_r(k-1) [M_n(k) = A322280(n,k)], m_n(k) = M_n(k) -Sum_{r=1..n-1} C(n-1,r-1)*m_r(k)*M_{n-r}(k) [m_n(k) = A322279(n,k)], f_n(k) = Sum_{r=1..k} (-1)^(k-r)*C(k,r)*m_n(r). This sequence gives a(n) = f_n(n). - Sean A. Irvine, May 29 2013, edited Andrew Howroyd, Dec 03 2018
The above formula is referenced by sequences A002027-A002030, A002031. - Andrew Howroyd, Dec 03 2018

More terms from Sean A. Irvine, May 29 2013
Name clarified by Andrew Howroyd, Dec 03 2018
a(0)=1 prepended by Andrew Howroyd, Jan 05 2024

A002027 Number of connected graphs on n labeled nodes, each node being colored with one of 2 colors, such that no edge joins nodes of the same color.

Original entry on oeis.org

1, 2, 2, 6, 38, 390, 6062, 134526, 4172198, 178449270, 10508108222, 853219059726, 95965963939958, 15015789392011590, 3282145108526132942, 1005193051984479922206, 432437051675617901246918, 261774334771663762228012950, 223306437526333657726283273822

Offset: 0

N. J. A. Sloane

nonn

a(n) is the number of connected labeled graphs with n 2-colored nodes where black nodes are only connected to white nodes and vice versa. - Geoffrey Critzer, Sep 05 2013

R. C. Read, personal communication.
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).

Andrew Howroyd, Table of n, a(n) for n = 0..50
R. C. Read, E. M. Wright, Colored graphs: A correction and extension, Canad. J. Math. 22 1970 594-596.

Column k=2 of A322279.
Essentially the same as A002031.
Cf. A002032.

nn=10;f[x_]:=Sum[Sum[Binomial[n,k]2^(k(n-k)),{k,0,n}]x^n/n!,{n,0,nn}];Range[0,nn]!CoefficientList[Series[Log[f[x]]+1,{x,0,nn}],x] (* Geoffrey Critzer, Sep 05 2013 *)

seq(n)={Vec(serlaplace(1 + log(serconvol(sum(j=0, n, x^j*2^binomial(j, 2)) + O(x*x^n), (sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n))^2))))} \\ Andrew Howroyd, Dec 03 2018

a(n) = m_n(2) using the functions defined in A002032. - Sean A. Irvine, May 29 2013
E.g.f.: log(A(x))+1 where A(x) is the e.g.f. for A047863. - Geoffrey Critzer, Sep 05 2013
Logarithmic transform of A047863. - Andrew Howroyd, Dec 03 2018

Corrected and extended by Sean A. Irvine, May 29 2013

Name clarified by Andrew Howroyd, Dec 03 2018

A274805 The logarithmic transform of sigma(n).

Original entry on oeis.org

1, 2, -3, -6, 45, 11, -1372, 4298, 59244, -573463, -2432023, 75984243, -136498141, -10881169822, 100704750342, 1514280063802, -36086469752977, -102642110690866, 11883894518252419, -77863424962770751, -3705485804176583500, 71306510264347489177

Offset: 1

Johannes W. Meijer, Jul 27 2016

sign

The logarithmic transform [LOG] transforms an input sequence b(n) into the output sequence a(n). The LOG transform is the inverse of the exponential transform [EXP], see the Weisstein link and the Sloane and Plouffe reference. This relation goes by the name of Riddell’s formula. For information about the EXP transform see A274804. The logarithmic transform is related to the inverse multinomial transform, see A274844 and the first formula.
The definition of the LOG transform, see the second formula, shows that n >= 1. To preserve the identity EXP[LOG[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the LOG transform, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the cumulant expansion numbers A127671 appear.
We observe that the logarithmic transform leaves the value of a(0) undefined.
The Maple programs can be used to generate the logarithmic transform of a sequence. The first program uses a formula found by Alois P. Heinz, see A001187 and the first formula. The second program uses the definition of the logarithmic transform, see the Weisstein link and the second formula. The third program uses information about the inverse of the logarithmic transform, see A274804.

			Some a(n) formulas, see A127671:
a(0) = undefined
a(1) = 1*x(1)
a(2) = 1*x(2) - x(1)^2
a(3) = 1*x(3) - 3*x(1)*x(2) + 2*x(1)^3
a(4) = 1*x(4) - 4*x(1)*x(3) - 3*x(2)^2 + 12*x(1)^2*x(2) - 6*x(1)^4
a(5) = 1*x(5) - 5*x(1)*x(4) - 10*x(2)*x(3) + 20*x(1)^2*x(3) + 30*x(1)*x(2)^2 - 60*x(1)^3*x(2) + 24*x(1)^5

Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973.
Robert James Riddell, Contributions to the theory of condensation, Dissertation, University of Michigan, Ann Arbor, 1951.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

Alois P. Heinz, Table of n, a(n) for n = 1..451
M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Logarithmic Transform.

Some LOG transform pairs are, n >= 1: A006125(n-1) and A033678(n); A006125(n) and A001187(n); A006125(n+1) and A062740(n); A000045(n) and A112005(n); A000045(n+1) and A007553(n); A000040(n) and A007447(n); A000051(n) and (-1)*A263968(n-1); A002416(n) and A062738(n); A000290(n) and A033464(n-1); A029725(n-1) and A116652(n-1); A052332(n) and A002031(n+1); A027641(n)/A027642(n) and (-1)*A060054(n+1)/(A075180(n-1).
Cf. A274804, A274844, A127671, A000203.
Cf. A112005, A007553, A062740, A007447, A062738, A033464, A116652, A002031, A003704, A003707, A155585, A000142, A226968.

nmax:=22: with(numtheory): b := proc(n): sigma(n) end: a:= proc(n) option remember; b(n) - add(k*binomial(n, k)*b(n-k)*a(k), k=1..n-1)/n: end: seq(a(n), n=1..nmax); # End first LOG program.
nmax:=22: with(numtheory): b := proc(n): sigma(n) end: t1 := log(1 + add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n=1..nmax); # End second LOG program.
nmax:=22: with(numtheory): b := proc(n): sigma(n) end: f := series(exp(add(r(n)*x^n/n!, n=1..nmax+1)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(1):=b(1): r(1):= b(1): for n from 2 to nmax+1 do r(n) := solve(d(n)-b(n), r(n)): a(n):=r(n): od: seq(a(n), n=1..nmax); # End third LOG program.

a[1] = 1; a[n_] := a[n] = DivisorSigma[1, n] - Sum[k*Binomial[n, k] * DivisorSigma[1, n-k]*a[k], {k, 1, n-1}]/n; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Feb 27 2017 *)

N=33; x='x+O('x^N); Vec(serlaplace(log(1+sum(n=1,N,sigma(n)*x^n/n!)))) \\ Joerg Arndt, Feb 27 2017

a(n) = b(n) - Sum_{k = 1..n-1}((k*binomial(n, k)*b(n-k)*a(k))/n), n >= 1, with b(n) = A000203(n) = sigma(n).

E.g.f. log(1+ Sum_{n >= 1}(b(n)*x^n/n!)), n >= 1, with b(n) = A000203(n) = sigma(n).

A342501 T(n,k) is the number of connected labeled posets with n elements and rank k: triangle read by rows.

Original entry on oeis.org

1, 0, 2, 0, 6, 6, 0, 38, 84, 24, 0, 390, 1710, 840, 120, 0, 6062, 49740, 36840, 8280, 720, 0, 134526, 2050566, 2184000, 646800, 85680, 5040, 0, 4172198, 118645044, 177549624, 65313360, 10735200, 947520, 40320

Offset: 1

R. J. Mathar, Mar 14 2021

This is a variant of A342587 admitting only connected posets.

			The table starts in row n=1 and shows ranks k>=0:
1: 1
2: 0 2
3: 0 6 6
4: 0 38 84 24
5: 0 390 1710 840 120
6: 0 6062 49740 36840 8280 720
7: 0 134526 2050566 2184000 646800 85680 5040
8: 0 4172198 118645044 177549624 65313360 10735200 947520 40320

Index entries for sequences related to posets

Cf. A001927 (row sums), A000142 (diagonal), A002031/A002027 (rank 1), A342500 (unlabeled).

A002028 Number of connected graphs on n labeled nodes, each node being colored with one of 3 colors, such that no edge joins nodes of the same color.

Original entry on oeis.org

1, 3, 6, 42, 618, 15990, 668526, 43558242, 4373213298, 677307561630, 162826875512646, 61183069270120842, 36134310487980825258, 33673533885068169649830, 49646105434209446798290206, 116002075479856331220877149042, 430053223599741677879550609246498, 2531493110297317758855120762121050990

Offset: 0

N. J. A. Sloane

nonn

R. C. Read, personal communication.
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).

Andrew Howroyd, Table of n, a(n) for n = 0..50
R. C. Read, E. M. Wright, Colored graphs: A correction and extension, Canad. J. Math. 22 1970 594-596.

Column k=3 of A322279.

Cf. A002031, A002032.

f[{k_, r_, m_}]:= Binomial[m+r+k, k] Binomial[m+r, r] 2^(k r +k m + r m);
  a = Sum[Total[Map[f, Compositions[n, 3]]] x^n/n!, {n, 0, 20}];
  Range[0, 20]! CoefficientList[Series[Log[a]+1, {x, 0, 20}], x] (* Geoffrey Critzer, Jun 02 2011 *)

seq(n)={Vec(serlaplace(1 + log(serconvol(sum(j=0, n, x^j*2^binomial(j, 2)) + O(x*x^n), (sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n))^3))))} \\ Andrew Howroyd, Dec 03 2018

E.g.f.: log(A(x))+1 where A(x) is the e.g.f. for A191371. - Geoffrey Critzer, Jun 02 2011
a(n) = m_n(3) using the functions defined in A002032. - Sean A. Irvine, May 29 2013
Logarithmic transform of A191371. - Andrew Howroyd, Dec 03 2018

Showing 1-10 of 15 results. Next

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A047863 Number of labeled graphs with 2-colored nodes where black nodes are only connected to white nodes and vice versa.

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A001831 Number of labeled graded partially ordered sets with n elements of height at most 1.

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A001832 Number of labeled connected bipartite graphs on n nodes.

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A052332 Number of labeled digraphs where every node has indegree 0 or outdegree 0 and no isolated nodes.

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A007776 Number of connected posets with n elements of height 1.

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A002032 Number of n-colored connected graphs on n labeled nodes.

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