A003027 -id:A003027 - cp's OEIS frontend

A361591 Triangle read by rows: T(n,k) is the number of weakly connected simple digraphs on n labeled nodes with k strongly connected components.

1, 0, 1, 0, 1, 2, 0, 18, 18, 18, 0, 1606, 1098, 684, 446, 0, 565080, 263580, 116370, 55620, 26430, 0, 734774776, 225806940, 68822910, 24578010, 9729090, 3596762, 0, 3523091615568, 680637057912, 136498491360, 34626926250, 10819771830, 3694824126, 1111506858

Offset: 0

Andrew Howroyd, May 04 2023

			Triangle begins:
  1;
  0,         1;
  0,         1,         2;
  0,        18,        18,       18;
  0,      1606,      1098,      684,      446;
  0,    565080,    263580,   116370,    55620,   26430;
  0, 734774776, 225806940, 68822910, 24578010, 9729090, 3596762;
  ...

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

Column k=1 is A003030.
Main diagonal is A082402.
Row sums are A003027.
The unlabeled version is A361587.
Cf. A189898, A361455.

\\ Uses functions defined in A361455.
T(n)={my(e=2); [Vecrev(p) | p<-Vec(serlaplace(1 + log(U(e, 1/G(e, exp(y*log(U(e, 1/G(e, DigraphEgf(n, e))))))))))]}
{ my(A=T(6)); for(i=1, #A, print(A[i])) }

A054914 Number of labeled connected digraphs with n nodes such that complement is also connected.

Original entry on oeis.org

1, 2, 44, 3572, 1005584, 1060875152, 4382913876704, 71987098738435232, 4721068803628864289024, 1237845578934919489219757312, 1298046978912816702510086132201984, 5444486716626952189940499391640815580672, 91343710775311761525117954724021374685703481344

Offset: 1

N. J. A. Sloane, May 23 2000

Alois P. Heinz, Table of n, a(n) for n = 1..50
V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A003027, A053763.

m:=30;
f:= func< x | (&+[2^(n*(n-1))*x^n/Factorial(n): n in [0..m+3]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( 1 + 2*Log(f(x)) - f(x) ))); // G. C. Greubel, Apr 28 2023

b:= n-> 2^(n^2-n):
g:= proc(n) option remember; local k; `if`(n=0, 1,
      b(n)- add(k*binomial(n,k) *b(n-k)*g(k), k=1..n-1)/n)
    end:
a:= n-> 2*g(n)-b(n):
seq (a(n), n=1..20);  # Alois P. Heinz, Oct 21 2012

nn=20; g=Sum[2^(2Binomial[n,2])x^n/n!,{n,0,nn}];
Drop[Range[0,nn]!CoefficientList[Series[2(Log[g]+1)-g,{x,0,nn}],x],1]  (* Geoffrey Critzer, Oct 21 2012 *)

m=30
def f(x): return sum(2^(n*(n-1))*x^n/factorial(n) for n in range(m+4))
def A054914_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 2 + 2*log(f(x)) - f(x) ).egf_to_ogf().list()
a=A054914_list(40); a[1:] # G. C. Greubel, Apr 28 2023

a(n) = 2*A003027(n) - A053763(n).

More terms from Vladeta Jovovic, Jul 17 2000

A177781 E.g.f. satisfies: L(x) = xSum_{n>=0} 3^n/n!Product_{k=0..n-1} L(4^k*x).

Original entry on oeis.org

1, 6, 162, 15336, 5135400, 6403850928, 30733361357328, 576178771105452672, 42495458789243292762240, 12378928091101498820594407680, 14278666564505879853034906179788544

Offset: 1

Paul D. Hanna, May 20 2010

nonn

More generally, we have the following conjecture.
Define the series E(,) and L(,) by:
. E(x,q) = Sum_{n>=0} q^(n(n-1)/2)*x^n/n!,
. L(x,q) = x*d/dx log(E(x,q)) = x*E'(x,q)/E(x,q),
then L(x,q) satisfies:
. L(x,q) = x*Sum_{n>=0} (q-1)^n/n! * Product_{k=0..n-1} L(q^k*x,q),
. 1/E(x,q) = Sum_{n>=0} (-1)^n/n! * Product_{k=0..n-1} L(q^k*x,q).
...
Explicitly, L(x,q) = [Sum_{n>=1} q^(n(n-1)/2)*x^n/(n-1)! ]/[Sum_{n>=0} q^(n(n-1)/2)*x^n/n! ]. - Paul D. Hanna, Aug 31 2010

			E.g.f.: L(x) = x + 6*x^2/2! + 162*x^3/3! + 15336*x^4/4! + 5135400*x^5/5! + ... + n*A003027(n)*x^n/n! + ...
Given the related expansions:
. E(x) = 1 + x + 4*x^2/2! +64*x^3/3! +4096*x^4/4! +1048576*x^5/5! + ...
. log(E(x)) = x + 3*x^2/2! +54*x^3/3! +3834*x^4/4! +1027080*x^5/5! + ... + A003027(n)*x^n/n! + ...
then L(x) satisfies:
. L(x)/x = 1 + 3*L(x) + 3^2*L(x)L(4x)/2! + 3^3*L(x)L(4x)L(16x)/3! + 3^4*L(x)L(4x)L(16x)L(64x)/4! + ...
. 1/E(x) = 1 - L(x) + L(x)L(4x)/2! - L(x)L(4x)L(16x)/3! + L(x)L(4x)L(16x)L(64x)/4! -+ ...

Cf. A003027, A177777, A177780.

{a(n,q=4)=local(Lq=x+x^2);for(i=1,n,Lq=x*sum(m=0,n,(q-1)^m/m!*prod(k=0,m-1,subst(Lq,x,q^k*x+x*O(x^n)))));n!*polcoeff(Lq,n)}

{a(n,q=4)=n!*polcoeff(sum(m=1,n,q^(m*(m-1)/2)*x^m/(m-1)!)/sum(m=0,n,q^(m*(m-1)/2)*x^m/m!+x*O(x^n)),n)} \\ Paul D. Hanna, Aug 31 2010

a(n) = n*A003027(n), where A003027(n) is the number of weakly connected digraphs with n nodes.
Define the series E(x) and L(x) by:
. E(x) = Sum_{n>=0} 4^(n(n-1)/2)*x^n/n!,
. L(x) = x*d/dx log(E(x)) = x*E'(x)/E(x),
then L(x) satisfies:
. L(x) = x*Sum_{n>=0} 3^n/n! * Product_{k=0..n-1} L(4^k*x),
. 1/E(x) = Sum_{n>=0} (-1)^n/n! * Product_{k=0..n-1} L(4^k*x).
...
E.g.f.: L(x) = [Sum_{n>=1} 4^(n(n-1)/2)*x^n/(n-1)! ]/[Sum_{n>=0} 4^(n(n-1)/2)*x^n/n! ]. - Paul D. Hanna, Aug 31 2010

A308458 Expansion of e.g.f. log(Sum_{k>=0} k^binomial(k,2) * x^k / k!).

Original entry on oeis.org

1, 1, 23, 3994, 9745169, 470126386536, 558542572785461515, 19342808645467142112096240, 22528399370853856386499346950471953, 999999999774716004550606847948627702867525440, 1890591424701781041871514584507296209311760279398415565711

Offset: 1

Seiichi Manyama, May 27 2019

nonn

			E.g.f.: x + x^2/2! + 23*x^3/3! + 3994*x^4/4! + 9745169*x^5/5! + 470126386536*x^6/6! + 558542572785461515*x^7/7! + ... .

Seiichi Manyama, Table of n, a(n) for n = 1..36

Cf. A001187, A001865, A003027, A054941.

N=20; x='x+O('x^N); Vec(serlaplace(log(sum(k=0, N, k^binomial(k, 2)*x^k/k!))))

A308460 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. log(Sum_{j>=0} k^binomial(j,2) * x^j/j!).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 20, 38, 0, 1, 4, 54, 624, 728, 0, 1, 5, 112, 3834, 55248, 26704, 0, 1, 6, 200, 15104, 1027080, 13982208, 1866256, 0, 1, 7, 324, 45750, 9684224, 1067308488, 10358360640, 251548592, 0, 1, 8, 490, 116208, 60225000, 30458183680, 4390480193904, 22792648882176, 66296291072, 0

Offset: 1

Seiichi Manyama, May 28 2019

			Square array begins:
   1,     1,        1,          1,           1, ...
   0,     1,        2,          3,           4, ...
   0,     4,       20,         54,         112, ...
   0,    38,      624,       3834,       15104, ...
   0,   728,    55248,    1027080,     9684224, ...
   0, 26704, 13982208, 1067308488, 30458183680, ...

Seiichi Manyama, Antidiagonals n = 1..50, flattened

Columns k=1..4 give A000007(n-1), A001187, A054941, A003027.

A127911 Number of nonisomorphic partial functional graphs with n points which are not functional graphs.

Original entry on oeis.org

0, 1, 3, 9, 26, 74, 208, 586, 1647, 4646, 13135, 37247, 105896, 301880, 862498, 2469480, 7083690, 20353886, 58571805, 168780848, 486958481, 1406524978, 4066735979, 11769294050, 34090034328, 98820719105, 286672555274

Offset: 0

Jonathan Vos Post, Feb 06 2007

Partial functional graphs (digraphs) with at least one node of outdegree = 0.

			a(0) = 0 because the null graph is trivially both partial functional and functional.
a(1) = 1 because there are two partial functional graphs on one point: the point with, or without, a loop; the point with loop is the identity function, but without a loop the naked point is the unique merely partial functional case.
a(2) = 3 because there are A126285(2) enumerates the 6 partial functional graphs on 2 points, of which 3 are functional, 6 - 3 = 3.
a(3) = A126285(3) - A001372(3) = 16 - 7 = 9.
a(4) = 45 - 19 = 26.
a(5) = 121 - 47 = 74.
a(6) = 338 - 130 = 208.
a(7) = 929 - 343 = 586.
a(8) = 2598 - 951 = 1647.
a(9) = 7261 - 2615 = 4646.
a(10) = 20453 - 7318 = 13135.

S. Skiena, "Functional Graphs." Section 4.5.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 164-165, 1990.

Eric Weisstein's World of Mathematics, Functional Graph.

Cf. A000081, A000273, A001372, A002861, A003027, A003085, A062738, A116950, A126285, A127909-A127915.

a(n) = A126285(n) - A001372(n).

Euler transform of (A002861 + A000081) - Euler transform of A002861.

A127912 Number of nonisomorphic disconnected mappings (or mapping patterns) from n points to themselves; number of disconnected endofunctions.

Original entry on oeis.org

0, 1, 3, 10, 27, 79, 218, 622, 1753, 5007, 14274, 40954, 117548, 338485, 975721, 2817871, 8146510, 23581381, 68322672, 198138512, 575058726, 1670250623, 4854444560, 14117859226, 41081418963, 119606139728

Offset: 0

Jonathan Vos Post, Feb 06 2007

Number of endofunctions on n points whose functional digraphs (with loops allowed) are nontrivially the directed sum of two or more digraphs of endofunctions.

			a(0) = 0, as the null digraph is formally neither connected nor disconnected.
a(1) = 0, as the unique endofunction on one point is the identity function on one value and is connected.
a(2) = 1, as there are 3 endofunctions on two points, two of which are "prime endofunctions" and one of which is the direct sum of two copies of the unique endofunction on one point, namely two points-with-loops, or the identity function on two values; 3 - 2 = 1.
a(3) = A001372(3) - A002861(3) = 7 - 4 = 3.
a(4) = A001372(4) - A002861(4) = 19 - 9 = 10.
a(5) = A001372(5) - A002861(5) = 47 - 20 = 27.
a(6) = 130 - 51 = 79.
a(7) = 343 - 125 = 218.
a(8) = 951 - 329 = 622.
a(9) = 2615 - 862 = 1753.
a(10) = 7318 - 2311 = 5007.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.6.
R. A. Fisher, Contributions to Mathematical Statistics, Wiley, 1950, 41.399 and 41.401.

N. G. de Bruijn and D. A. Klarner, Multisets of aperiodic cycles, SIAM J. Algebraic Discrete Methods 3 (1982), no. 3, 359--368. MR0666861(84i:05008).
Eric Weisstein's World of Mathematics, Functional Graph.

Cf. A000081, A000273, A001372, A002861, A003027, A003085, A062738, A116950, A126285, A127909-A127915.

a(n) = A001372(n) - A002861(n).

cp's OEIS Frontend

A361591 Triangle read by rows: T(n,k) is the number of weakly connected simple digraphs on n labeled nodes with k strongly connected components.

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A054914 Number of labeled connected digraphs with n nodes such that complement is also connected.

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A177781 E.g.f. satisfies: L(x) = x*Sum_{n>=0} 3^n/n!*Product_{k=0..n-1} L(4^k*x).

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A308458 Expansion of e.g.f. log(Sum_{k>=0} k^binomial(k,2) * x^k / k!).

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A308460 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. log(Sum_{j>=0} k^binomial(j,2) * x^j/j!).

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A127911 Number of nonisomorphic partial functional graphs with n points which are not functional graphs.

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A127912 Number of nonisomorphic disconnected mappings (or mapping patterns) from n points to themselves; number of disconnected endofunctions.

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A177781 E.g.f. satisfies: L(x) = xSum_{n>=0} 3^n/n!Product_{k=0..n-1} L(4^k*x).