A052332 -id:A052332 - 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

A002031 Number of labeled connected digraphs on n nodes where every node has indegree 0 or outdegree 0 and no isolated nodes.

Original entry on oeis.org

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

Offset: 2

N. J. A. Sloane

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

In- or outdegree zero implies loops are not admitted. Multi-arcs are not admitted. - R. J. Mathar, Nov 18 2023

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).

Alois P. Heinz, Table of n, a(n) for n = 2..100
R. C. Read, E. M. Wright, Colored graphs: A correction and extension, Canad. J. Math. 22 1970 594-596.
N. J. A. Sloane, Transforms

Cf. A001831, A001832, A002032, A047863, A052332, A007776 (unlabeled case). Essentially the same as A002027.

logtr:= proc(p) local b; b:=proc(n) option remember; local k; if n=0 then 1 else p(n)- add(k *binomial(n,k) *p(n-k) *b(k), k=1..n-1)/n fi end end: digr:= n-> add(binomial(n,k) *(2^k-2)^(n-k), k=0..n): a:= logtr(digr): seq(a(n), n=2..25);  # Alois P. Heinz, Sep 14 2008

terms = 17; s = Log[Sum[Exp[(2^n - 2)*x]*(x^n/n!), {n, 0, terms+2}]] + O[x]^(terms+2); Drop[CoefficientList[s, x]*Range[0, terms+1]!, 2] (* Jean-François Alcover, Nov 08 2011, after Vladeta Jovovic, updated Jan 12 2018 *)

Logarithmic transform of A052332.
E.g.f.: log(Sum(exp((2^n-2)*x)*x^n/n!, n=0..infinity)). - Vladeta Jovovic, May 28 2004
a(n) = f(n,2) using functions defined in A002032. - Sean A. Irvine, May 29 2013

More terms, formula and new title from Christian G. Bower, Dec 15 1999

Corrected by Vladeta Jovovic, Apr 12 2003

A218695 Square array A(h,k) = (2^h-1)A(h,k-1) + Sum_{i=1..h-1} binomial(h,h-i)2^i*A(i,k-1), with A(1,k) = A(h,1) = 1; read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 25, 25, 1, 1, 79, 265, 79, 1, 1, 241, 2161, 2161, 241, 1, 1, 727, 16081, 41503, 16081, 727, 1, 1, 2185, 115465, 693601, 693601, 115465, 2185, 1, 1, 6559, 816985, 10924399, 24997921, 10924399, 816985, 6559, 1

Offset: 1

M. F. Hasler, Nov 04 2012

This symmetric table is defined in the Kreweras papers, used also in A223911. Its upper or lower triangular part equals A183109, which might provide a simpler formula.
Number of h X k binary matrices with no zero rows or columns. - Andrew Howroyd, Mar 29 2023
A(h,k) is the number of coverings of [h] by tuples (A_1,...,A_k) in P([h])^k with nonempty A_j, with P(.) denoting the power set. For the disjoint case see A019538. For tuples with "nonempty" omitted see A092477 and A329943 (amendment by Manfred Boergens, Jun 24 2024). - Manfred Boergens, May 26 2024

			Array A(h,k) begins:
=====================================================
h\k | 1   2      3        4         5           6 ...
----+------------------------------------------------
  1 | 1   1      1        1         1           1 ...
  2 | 1   7     25       79       241         727 ...
  3 | 1  25    265     2161     16081      115465 ...
  4 | 1  79   2161    41503    693601    10924399 ...
  5 | 1 241  16081   693601  24997921   831719761 ...
  6 | 1 727 115465 10924399 831719761 57366997447 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
G. Kreweras, Dénombrements systématiques de relations binaires externes, Math. Sci. Humaines 26 (1969) 5-15.
G. Kreweras, Dénombrement des ordres étagés, Discrete Math., 53 (1985), 147-149.

Columns 1..3 are A000012, A058481, A058482.
Main diagonal is A048291.
Cf. A019538, A056152 (unlabeled case), A052332, A092477, A183109, A223911, A329943.

c(h,k)={(h<2 || k<2) & return(1); sum(i=1,h-1,binomial(h,h-i)*2^i*c(i,k-1))+(2^h-1)*c(h,k-1)}
/* For better performance when h and k are large, insert the following memoization code before "sum(...)": cM=='cM & cM=matrix(h,k); my(s=matsize(cM));
s[1] >= h & s[2] >= k & cM[h,k] & return(cM[h,k]);
s[1]

A(m, n) = sum(k=0, m, (-1)^(m-k) * binomial(m, k) * (2^k-1)^n ) \\ Andrew Howroyd, Mar 29 2023

From Andrew Howroyd, Mar 29 2023: (Start)
A(h, k) = Sum_{i=0..h} (-1)^(h-i) * binomial(h, i) * (2^i-1)^k.
A052332(n) = Sum_{i=1..n-1} binomial(n,i)*A(i, n-i) for n > 0. (End)

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).

A361956 Triangle read by rows: T(n,k) is the number of labeled tiered posets with n elements and height k.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 50, 36, 24, 0, 1, 510, 510, 240, 120, 0, 1, 7682, 10620, 4800, 1800, 720, 0, 1, 161406, 312606, 136920, 47040, 15120, 5040, 0, 1, 4747010, 13439076, 5630184, 1678320, 493920, 141120, 40320, 0, 1, 194342910, 821218110, 319384800, 83963880, 21137760, 5594400, 1451520, 362880

Offset: 0

Andrew Howroyd, Apr 02 2023

A tiered poset is a partially ordered set in which every maximal chain has the same length.

			Triangle begins:
  1;
  0, 1;
  0, 1,      2;
  0, 1,      6,      6;
  0, 1,     50,     36,     24;
  0, 1,    510,    510,    240,   120;
  0, 1,   7682,  10620,   4800,  1800,   720;
  0, 1, 161406, 312606, 136920, 47040, 15120, 5040;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50).

Row sums are A223911.
Column k=2 is A052332.
Main diagonal is A000142.
The unlabeled version is A361957.
Cf. A222864, A361951.

S(M)={my(N=matrix(#M-1, #M-1, i, j, sum(k=1, i-j+1, (2^j-1)^k*M[i-j+1, k])/j!)); for(i=1, #N, for(j=1, i, N[i,j] -= sum(k=1, j-1, N[i-k, j-k]/k!))); N}
C(n)={my(M=matrix(n+1,n+1), R=M); M[1,1]=R[1,1]=1; for(h=1, n, M=S(M); for(i=h, n, R[i+1,h+1] = i!*vecsum(M[i-h+1,]))); R}
{ my(A=C(7)); for(i=1, #A, print(A[i, 1..i])) }

A122801 Number of labeled bipartite graphs on 2n vertices having equal parts and no isolated vertices.

Original entry on oeis.org

1, 1, 21, 2650, 1452605, 3149738046, 26503552820514, 868081172737564500, 111606080497500509325405, 56762846667123360827351083510, 114847831981827229530824587617895286, 927685362544629192461621864598358779955500, 29976424929810726580224613882836823991388901138994

Offset: 0

Max Alekseyev, Sep 11 2006

nonn

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

Cf. A122802, A048291, A052332, A001831, A002031, A047863, A001700.

{ A122801(n) = binomial(2*n-1,n) * sum(k=0, n, binomial(n, k) * (-1)^k * (2^(n-k)-1)^n ); }

For n>0, a(n) = A001700(n-1) * A048291(n) = A052332(2n) - A122802(2n).

Terms a(11) and beyond from Andrew Howroyd, Nov 07 2019

A122802 Number of labeled bipartite graphs on n vertices with no isolated vertices.

Original entry on oeis.org

1, 0, 1, 6, 29, 510, 5032, 161406, 3294405, 194342910, 7934652356, 881008805886, 71275547085536, 15178191426486270, 2434250064518832302, 1008694542117649154046, 321680912414994434144165, 262063364967549826752315390, 166681427053102507699172431372

Offset: 0

Max Alekseyev, Sep 11 2006

nonn

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

Cf. A122801, A052332, A001831, A002031, A047863.

a(n)={sum(k=0, n, binomial(n, k)*(2^k-2)^(n-k)) - if(n%2==0&&n>0, binomial(n-1, n/2)*sum(k=0, n/2, binomial(n/2, k)*(-1)^k*(2^(n/2-k)-1)^(n/2)))} \\ Andrew Howroyd, Nov 07 2019

a(2n+1) = A052332(2n+1); a(2n) = A052332(2n) - A122801(n) for n > 0.

Terms a(17) and beyond from Andrew Howroyd, Nov 07 2019

A120667 Number of n-node labeled bipartite graphs without isolated nodes.

Original entry on oeis.org

1, 0, 1, 3, 22, 225, 3421, 73668, 2222977, 93033615, 5393456986, 433396737873, 48429436851577, 7548123580987080, 1646092439020192801, 503469306031901522043, 216430661498688457821022, 130959358877474026010486145, 111687660283090149155082836341

Offset: 0

Vladeta Jovovic, Jun 23 2007

Alois P. Heinz, Table of n, a(n) for n = 0..60

Cf. A047864.

a:= n-> coeff (series (sqrt (add (exp (x*(2^k-2)) *x^k/k!, k=0..n)), x, n+1), x, n)*n!: seq (a(n), n=0..20);  # Alois P. Heinz, Sep 12 2008

E.g.f.: sqrt( e.g.f. for A052332 ) = sqrt(Sum_{n>=0} exp(x*(2^n-2)) * x^n/n!).

More terms from Alois P. Heinz, Sep 12 2008

A174122 Partial sums of A001831.

Original entry on oeis.org

1, 2, 5, 18, 105, 946, 12589, 240482, 6526289, 250119330, 13512676053, 1027978959346, 110155994874553, 16631509898085074, 3540687511804739837, 1063409634646294541250, 450894476653951603096737

Offset: 0

Jonathan Vos Post, Mar 08 2010

nonn

Partial sums of number of labeled graded partially ordered sets with n elements. The subsequence of primes in this partial sum begins: 2, 5, 12589.

Cf. A001831, A002031, A047863, A052332, A001833, A048194, A052296, A135753, A135754.

a(n) = SUM[i=0..n] A001831(i) = SUM[i=0..n] SUM[j=0..i] ((-1)^j*C(n,j)*A047863(j)).

Showing 1-10 of 10 results.

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

Python

SageMath

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A002031 Number of labeled connected digraphs on n nodes where every node has indegree 0 or outdegree 0 and no isolated nodes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A218695 Square array A(h,k) = (2^h-1)*A(h,k-1) + Sum_{i=1..h-1} binomial(h,h-i)*2^i*A(i,k-1), with A(1,k) = A(h,1) = 1; read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A274805 The logarithmic transform of sigma(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A361956 Triangle read by rows: T(n,k) is the number of labeled tiered posets with n elements and height k.

Views

Author

Keywords

Comments

A218695 Square array A(h,k) = (2^h-1)A(h,k-1) + Sum_{i=1..h-1} binomial(h,h-i)2^i*A(i,k-1), with A(1,k) = A(h,1) = 1; read by antidiagonals.