A001831 -id:A001831 - cp's OEIS frontend

A174122 Partial sums of A001831.

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)).

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

Original entry on oeis.org

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

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

A048194 Total number of split graphs (chordal + chordal complement) on n vertices.

Original entry on oeis.org

1, 2, 4, 9, 21, 56, 164, 557, 2223, 10766, 64956, 501696, 5067146, 67997750, 1224275498, 29733449510, 976520265678, 43425320764422, 2616632636247976, 213796933371366930, 23704270652844196754, 3569464106212250952762, 730647291666881838671052

Offset: 1

Gordon F. Royle

Also number of bipartite graphs with n vertices and no isolated vertices in distinguished bipartite block, up to isomorphism; so a(n) equals first differences of A049312. - Vladeta Jovovic, Jun 17 2000
All split graphs are perfect. - Falk Hüffner, Nov 29 2015
Inverse Euler transform gives A007776 with initial 1. - Andrew Howroyd, Oct 03 2018

Alois P. Heinz, Table of n, a(n) for n = 1..40
B. A. Chat, S. Pirzada, and A. Iványi, Recognition of split-graphic sequences, Acta Universitatis Sapientiae, Informatica, 6, 2 (2014) 252-286.
Karen L. Collins and Ann N. Trenk, Finding Balance: Split Graphs and Related Classes, arXiv:1706.03092 [math.CO], June 2017.
Karen L. Collins and Ann N. Trenk, Finding Balance: Split Graphs and Related Classes, Electron. J. Combin., 25 (2018), #P1.73.
S. Hougardy, Home Page.
S. Hougardy, Classes of perfect graphs, Discr. Math. 306 (2006), 2529-2571.
Vladeta Jovovic, Binary matrices up to row and column permutations.
Gordon F. Royle, Counting set covers and split graphs, J. Integer Seqs., Vol. 3 (2000), #00.2.6.
J. M. Troyka, Split graphs: combinatorial species and asymptotics, Electron. J. Combin., 26 (2019), #P2.42.
J. M. Troyka, Split graphs: combinatorial species and asymptotics, arXiv:1803.07248 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Split Graph.
Index entries for sequences related to posets

Cf. A007776, A048192, A048193, A049312, A055080, A263859.

Detlef Pauly remarks that this is the unlabeled analog of A001831.

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]];
a[d_] := Sum[A[n, d - n], {n, 0, d}] - Sum[A[n, d - n - 1], {n, 0, d - 1}];
Table[a[n], {n, 1, 25}] (* Jean-François Alcover, May 26 2019, after Alois P. Heinz in A049312 *)

a(n) = A049312(n) - A049312(n-1) (see the Collins and Trenk link, Thms. 5 and 15). - Justin M. Troyka, Oct 29 2018
a(n) ~ A049312(n) ~ (1/n!) * Sum_{k=0..n} binomial(n,k) * 2^(k(n-k)) (see the Troyka link, Thms. 3.7 and 3.10). - Justin M. Troyka, Oct 29 2018
a(n) = A263859(n,1) + 1. - Geoffrey Critzer, Feb 05 2024

A001833 Number of labeled graded partially ordered sets with n elements.

Original entry on oeis.org

1, 1, 3, 19, 219, 3991, 106623, 3964339, 199515459, 13399883551, 1197639892983, 143076298623259, 23053861370437659, 5062745845287855271, 1530139311543346178223, 641441466132460086890179, 375107113287994040621904819, 307244526491924695346004951151, 353511145615118063468292270299943

Offset: 0

N. J. A. Sloane

nonn

Here "graded" means that there exists a rank function rk from the poset to the integers such that whenever v covers w in the poset, we have rk(v) = rk(w) + 1. Note that this notion of grading is weaker than in sequence A006860, which counts posets in which all maximal chains have the same length.

			The poset on {a, b, c, d, e} defined by the relations a < b < c and d < e is counted by this sequence. (For example, one associated rank function is rk(a) = rk(d) = 0, rk(b) = rk(e) = 1 and rk(c) = 2.) However, the poset defined by the relations a < b < c and a < d < e < c is not graded and so not counted by this sequence.

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..100
David A. Klarner, The number of graded partially ordered sets, Journal of Combinatorial Theory, vol.6, no.1, pp.12-19, (January-1969).
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
Index entries for sequences related to posets

Row sums of A361951.
Graded posets with no chain of length 3 are counted by A001831.
Cf. A223911, A228551, A361920 (unlabeled version).

\\ C(n) is defined in A361951.
seq(n)={my(c=C(n)); Vec(serlaplace(c[n+1]/c[n]))} \\ Andrew Howroyd, Mar 31 2023

Corrected and edited by Joel B. Lewis, Mar 28 2011
a(7)-a(15) from Daniele P. Morelli, Aug 25 2013
a(16)-a(18) from Sean A. Irvine, Sep 25 2015

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

A052296 Triangle read by rows: T(n,k) = number of labeled digraphs with n nodes and k arcs and without directed paths of length >=2, with 0 <= k <= floor(n^2/4).

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 6, 1, 12, 36, 32, 6, 1, 20, 120, 280, 280, 120, 20, 1, 30, 300, 1320, 2910, 3492, 2400, 960, 210, 20, 1, 42, 630, 4480, 17220, 39144, 56294, 53760, 35070, 15680, 4662, 840, 70, 1, 56, 1176, 12320, 73220, 269136, 654304, 1108928, 1362900

Offset: 0

Vladeta Jovovic, Feb 08 2000

			  1;
  1;
  1,  2;
  1,  6,   6;
  1, 12,  36,   32,    6;
  1, 20, 120,  280,  280,  120,   20;
  1, 30, 300, 1320, 2910, 3492, 2400, 960, 210, 20;
  ...

Row sums give A001831.

Cf. A002378 (k=1), A083374 (k=2).

A052296 := proc(n,k)
    local x,l ;
    add(binomial(n,l)*((1+x)^l-1)^(n-l),l=0..n) ;
    expand(%) ;
    coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Mar 16 2021

G.f. for n-th row: Sum_{k=0..n} binomial(n, k)*((1+x)^k-1)^(n-k). - Vladeta Jovovic, Apr 04 2003
E.g.f.: Sum_{n>=0} exp(y*((1+x)^n-1))*y^n/n!. - Vladeta Jovovic, May 28 2004
T(n,3) = n*(n-1)*(n-2)*(n-3)*(n^2-3*n+4)/6, n>=4. - R. J. Mathar, Mar 16 2021

A135753 E.g.f.: A(x) = Sum_{n>=0} exp((3^n-1)/2x)x^n/n!.

Original entry on oeis.org

1, 1, 3, 16, 153, 2536, 72513, 3571156, 303033153, 44411895376, 11247688063233, 4933176144494236, 3746180187749948193, 4933259445571307491096, 11257237602638666745470913, 44566655569041016108120599556

Offset: 0

Paul D. Hanna, Nov 27 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..85

Cf. variants: A001831, A135754.

Flatten[{1,Table[Sum[Binomial[n,k]*((3^k-1)/2)^(n-k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jun 25 2013 *)

a(n)=sum(k=0,n,binomial(n,k)*((3^k-1)/2)^(n-k))

a(n)=n!*polcoeff(sum(k=0,n,exp((3^k-1)/2*x)*x^k/k!),n)

a(n) = Sum_{k=0..n} binomial(n,k)*[(3^k-1)/2]^(n-k).

a(n) ~ c * 3^(n^2/4)*2^((n+1)/2)/sqrt(Pi*n), where c = Sum_{k = -infinity..infinity} 2^k*3^(-k^2) = 1.8862156350800186... if n is even and c = Sum_{k = -infinity..infinity} 2^(k+1/2)*3^(-(k+1/2)^2) = 1.8865940733664341... if n is odd. - Vaclav Kotesovec, Jun 25 2013

A135754 E.g.f.: A(x) = Sum_{n>=0} exp((4^n-1)/3x)x^n/n!.

Original entry on oeis.org

1, 1, 3, 19, 239, 6091, 305023, 30818299, 6155906879, 2484667187371, 1989929726352863, 3221489148102557179, 10362312712649347408159, 67345216546226371822133611, 869978904614825017953532433663

Offset: 0

Paul D. Hanna, Nov 27 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..80

Cf. variants: A001831, A135753.

Flatten[{1,Table[Sum[Binomial[n,k]*((4^k-1)/3)^(n-k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jun 25 2013 *)

a(n)=sum(k=0,n,binomial(n,k)*((4^k-1)/3)^(n-k))

a(n)=n!*polcoeff(sum(k=0,n,exp((4^k-1)/3*x)*x^k/k!),n)

a(n) = Sum_{k=0..n} C(n,k)*[(4^k-1)/3]^(n-k).

a(n) ~ c * 2^(n^2/2+n+1/2)/(3^(n/2)*sqrt(Pi*n)), where c = Sum_{k = -infinity..infinity} 3^k*4^(-k^2) = 1.86902676808473931... if n is even and c = Sum_{k = -infinity..infinity} 3^(k+1/2)*4^(-(k+1/2)^2) = 1.87384213421283135... if n is odd. - Vaclav Kotesovec, Jun 25 2013

A361951 Triangle read by rows: T(n,k) is the number of labeled weakly graded (ranked) posets with n elements and rank k.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 12, 6, 0, 1, 86, 108, 24, 0, 1, 840, 2190, 840, 120, 0, 1, 11642, 55620, 31800, 6840, 720, 0, 1, 227892, 1858206, 1428000, 384720, 60480, 5040, 0, 1, 6285806, 82938828, 80529624, 24509520, 4626720, 584640, 40320

Offset: 0

Andrew Howroyd, Mar 31 2023

Here weakly graded means that there exists a rank function rk from the poset to the integers such that whenever v covers w in the poset, we have rk(v) = rk(w) + 1.

T(n,k) corresponds to a(k,n) = b(k,n) - b(k-1,n) in the Klarner reference. Figure 2 shows the posets of row n=4.

			Triangle begins:
  1;
  0, 1;
  0, 1,      2;
  0, 1,     12,       6;
  0, 1,     86,     108,      24;
  0, 1,    840,    2190,     840,    120;
  0, 1,  11642,   55620,   31800,   6840,   720;
  0, 1, 227892, 1858206, 1428000, 384720, 60480, 5040;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50).
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.
Wikipedia, Graded poset.

Row sums are A001833.
Column k=2 is A055531.
Partial row sums include A000007, A000012, A001831, A001832.
Main diagonal is A000142.
The unlabeled version is A361953.
Cf. A342587, A361950.

\\ Here C(n) gives columns of A361950 as vector of e.g.f.'s.
S(M)={matrix(#M, #M, i, j, sum(k=0,  i-j, 2^((j-1)*k)*M[i-j+1,k+1])/(j-1)! )}
C(n,m=n)={my(M=matrix(n+1, n+1), c=vector(m+1), A=O(x*x^n)); M[1, 1]=1; c[1]=1+A; for(h=1, m, M=S(M); c[h+1]=sum(i=0, n, vecsum(M[i+1, ])*x^i, A)); c}
T(n)={my(c=C(n), b=vector(n+1, h, c[h]/c[max(h-1,1)])); Mat(vector(n+1, h, Col(serlaplace(b[h]-if(h>1, b[h-1])), -n-1)))}
{ my(A=T(7)); for(n=1, #A, print(A[n, 1..n])) }

E.g.f. of column k >=2: C(k,x)/C(k-1,x) - C(k-1,x)/C(k-2,x) where C(k,x) is the e.g.f. of column k of A361950.

cp's OEIS Frontend

A174122 Partial sums of A001831.

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

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A048194 Total number of split graphs (chordal + chordal complement) on n vertices.

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A001833 Number of labeled graded partially ordered sets with n elements.

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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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A052296 Triangle read by rows: T(n,k) = number of labeled digraphs with n nodes and k arcs and without directed paths of length >=2, with 0 <= k <= floor(n^2/4).

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A135753 E.g.f.: A(x) = Sum_{n>=0} exp((3^n-1)/2*x)*x^n/n!.

A135753 E.g.f.: A(x) = Sum_{n>=0} exp((3^n-1)/2x)x^n/n!.

A135754 E.g.f.: A(x) = Sum_{n>=0} exp((4^n-1)/3x)x^n/n!.