A005993 -id:A005993 - cp's OEIS frontend

A106314 Triangle T(n,k) composed of the squares min(n,k)^2.

1, 1, 1, 1, 4, 1, 1, 4, 4, 1, 1, 4, 9, 4, 1, 1, 4, 9, 9, 4, 1, 1, 4, 9, 16, 9, 4, 1, 1, 4, 9, 16, 16, 9, 4, 1, 1, 4, 9, 16, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 25, 16, 9, 4, 1

Offset: 1

Gary W. Adamson, Apr 28 2005

			Replacing each term in A003983 by its square, we get:
{1},
{1, 1},
{1, 4, 1},
{1, 4, 4, 1},
{1, 4, 9, 4, 1},
{1, 4, 9, 9, 4, 1},
{1, 4, 9, 16, 9, 4, 1},
{1, 4, 9, 16, 16, 9, 4, 1},
{1, 4, 9, 16, 25, 16, 9, 4, 1},
{1, 4, 9, 16, 25, 25, 16, 9, 4, 1},
{1, 4, 9, 16, 25, 36, 25, 16, 9, 4, 1}

Cf. A003983, A106314, A005993 (row sums).

Clear[p, n, i];
p[x_, n_] = Sum[x^i*If[i ==Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= Floor[n/2], 2*i + 1, -(2*(n - i) + 1)]], {i, 0, n}]/(1 - x);
Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 1, 11}];
Flatten[%]

T(n,k) = A003983(n,k)^2.

Additional comments from Roger L. Bagula and Gary W. Adamson, Apr 02 2009

A136564 Array read by rows: T(n,k) is the number of directed multigraphs with loops with n arcs, k vertices, and no vertex of degree 0.

Original entry on oeis.org

1, 1, 1, 5, 4, 1, 1, 9, 21, 16, 4, 1, 1, 18, 71, 108, 71, 22, 4, 1, 1, 27, 194, 491, 557, 326, 101, 22, 4, 1, 1, 43, 476, 1903, 3353, 3062, 1587, 497, 111, 22, 4, 1, 1, 59, 1030, 6298, 16644, 22352, 17035, 7982, 2433, 555, 111, 22, 4, 1, 1, 84, 2095, 18823, 72064

Offset: 1

Benoit Jubin, Apr 14 2008

Length of the n^th row: 2n.

			1, 1;
1, 5, 4, 1;
1, 9, 21, 16, 4, 1;
1, 18, 71, 108, 71, 22, 4, 1;
1, 27, 194, 491, 557, 326, 101, 22, 4, 1;
1, 43, 476, 1903, 3353, 3062, 1587, 497, 111, 22, 4, 1;
1, 59, 1030, 6298, 16644, 22352, 17035, 7982, 2433, 555, 111, 22, 4, 1;

Row sums: A052171. Partial row sums: A138107.

Sums of the first m entries of each row: A005993 (m=2), A050927 (m=3), A050929 (m=4).

T(n,1) = 1 if n > 0.
T(n,2n) = 1 if n > 0.
T(n,2n-1) = 4 if n >= 2.
T(n,2n-k) = A144047(k) for n large enough (conjecturally, n >= 2k is enough).
T(n,2) = (n^3 + 6*n^2 + 11*n - 6)/12 + ((n+2)/4)[n even]. (the bracket means that the second term is added if and only if n is even). - Benoit Jubin, Mar 31 2012

More terms from Benoit Jubin and Vladeta Jovovic, Sep 08 2008

A248011 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing three 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 6, 6, 2, 6, 14, 27, 14, 6, 10, 32, 60, 60, 32, 10, 19, 55, 129, 140, 129, 55, 19, 28, 94, 218, 294, 294, 218, 94, 28, 44, 140, 363, 506, 608, 506, 363, 140, 44, 60, 208, 536, 832, 1038, 1038, 832, 536, 208, 60, 85, 285, 785, 1240, 1695

Offset: 1

Christopher Hunt Gribble, Sep 29 2014

			T(n,k) for 1<=n<=9 and 1<=k<=9 is:
   k    1     2     3     4     5     6     7     8     9 ...
n
1       0     0     1     2     6    10    19    28    44
2       0     1     6    14    32    55    94   140   208
3       1     6    27    60   129   218   363   536   785
4       2    14    60   140   294   506   832  1240  1802
5       6    32   129   294   608  1038  1695  2516  3642
6      10    55   218   506  1038  1785  2902  4324  6242
7      19    94   363   832  1695  2902  4703  6992 10075
8      28   140   536  1240  2516  4324  6992 10416 14988
9      44   208   785  1802  3642  6242 10075 14988 21544

Christopher Hunt Gribble, Table of n, a(n) for n = 1..9870

Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A243866, A006918, A244306, A244307, A248016, A248059, A248060, A248017, A248027.

b := proc (n::integer, k::integer)::integer;
(4*k^3*n^3 - 12*k^2*n^2 + 2*k^3 + 6*k^2*n + 6*k*n^2 + 2*n^3 - 12*k^2 + 11*k*n - 12*n^2 + 4*k + 4*n - 3 - (2*k^3 + 6*k^2*n - 12*k^2 + 3*k*n + 4*k - 3)*(-1)^n - (6*k*n^2 + 2*n^3 + 3*k*n - 12*n^2 + 4*n - 3)*(-1)^k + (3*k*n - 3)*(-1)^k*(-1)^n)*(1/96);
end proc;
f := seq(seq(b(n, k - n + 1), n = 1 .. k), k = 1 .. 140);

Empirically,
T(n,k) = (4*k^3*n^3 - 12*k^2*n^2 + 2*k^3 + 6*k^2*n + 6*k*n^2 + 2*n^3 - 12*k^2 + 11*k*n - 12*n^2 + 4*k + 4*n - 3 - (2*k^3 + 6*k^2*n - 12*k^2 + 3*k*n + 4*k - 3)*(-1)^n - (6*k*n^2 + 2*n^3 + 3*k*n - 12*n^2 + 4*n - 3)*(-1)^k + (3*k*n - 3)*(-1)^k*(-1)^n)/96;
T(1,k) = A005993(k-3) = (k-1)*(2*(k-2)*k + 3*(1-(-1)^k))/24;
T(2,k) = A225972(k) = (k-1)*(2*k*(2*k-1) + 3*(1-(-1)^k))/12;
T(2,k) - T(1,k) = A199771(k-1) and A212561(k) = (k-1)*(6*k^2 + 3*(1-(-1)^k))/24.

Terms corrected and extended by Christopher Hunt Gribble, Apr 01 2015

A248059 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing four 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 6, 6, 1, 3, 22, 39, 22, 3, 9, 60, 139, 139, 60, 9, 19, 135, 371, 476, 371, 135, 19, 38, 266, 813, 1253, 1253, 813, 266, 38, 66, 476, 1574, 2706, 3254, 2706, 1574, 476, 66, 110, 792, 2770, 5199, 6969, 6969, 5199, 2770, 792, 110, 170, 1245

Offset: 1

Christopher Hunt Gribble, Sep 30 2014

			T(n,k) for 1<=n<=9 and 1<=k<=9 is:
   k    1      2      3      4      5      6      7      8       9 ...
n
1       0      0      0      1      3      9     19     38      66
2       0      1      6     22     60    135    266    476     792
3       0      6     39    139    371    813   1574   2770    4554
4       1     22    139    476   1253   2706   5199   9080   14857
5       3     60    371   1253   3254   6969  13294  23102   37637
6       9    135    813   2706   6969  14841  28197  48852   79401
7      19    266   1574   5199  13294  28197  53381  92266  149645
8      38    476   2770   9080  23102  48852  92266 159216  257878
9      66    792   4554  14857  37637  79401 149645 257878  417156

Christopher Hunt Gribble, Table of n, a(n) for n = 1..9870

Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A243866, A006918, A244306, A244307, A248011, A248016, A248060, A248017, A248027.

b := proc (n::integer, k::integer)::integer;
(4*k^4*n^4 - 24*k^3*n^3 + 2*k^4 + 12*k^3*n + 80*k^2*n^2 + 12*k*n^3 + 2*n^4 - 24*k^3 - 24*k^2*n - 24*k*n^2 - 24*n^3 + 40*k^2 - 102*k*n + 40*n^2 + 9 + (- 2*k^4 - 12*k^3*n + 24*k^3 + 24*k^2*n - 40*k^2 + 6*k*n - 9)*(-1)^n + (- 12*k*n^3 - 2*n^4 + 24*k*n^2 + 24*n^3 + 6*k*n - 40*n^2 - 9)*(-1)^k + (- 6*k*n + 9)*(-1)^k*(-1)^n)/384
end proc;
seq(seq(b(n, k-n+1), n = 1 .. k), k = 1 .. 140);

Empirically,
T(n,k) = (4*k^4*n^4 - 24*k^3*n^3 + 2*k^4 + 12*k^3*n + 80*k^2*n^2 + 12*k*n^3 + 2*n^4 - 24*k^3 - 24*k^2*n - 24*k*n^2 - 24*n^3 + 40*k^2 - 102*k*n + 40*n^2 + 9 + (- 2*k^4 - 12*k^3*n + 24*k^3 + 24*k^2*n - 40*k^2 + 6*k*n - 9)*(-1)^n + (- 12*k*n^3 - 2*n^4 + 24*k*n^2 + 24*n^3 + 6*k*n - 40*n^2 - 9)*(-1)^k + (- 6*k*n + 9)*(-1)^k*(-1)^n)/384;
T(1,k) = sum(A005993(i-4),i=1,k)
       = sum((i-2)*(2*(i-3)*(i-1) + 3*(1-(-1)^(i-1)))/24, i=1,k);
T(2,k) = A071239(k-1) = (k-1)*k*((k-1)^2+2)/6.

Terms corrected and extended by Christopher Hunt Gribble, Apr 06 2015

A050927 Number of directed multigraphs with loops on 3 nodes with n arcs.

Original entry on oeis.org

1, 2, 10, 31, 90, 222, 520, 1090, 2180, 4090, 7356, 12660, 21105, 34020, 53460, 81891, 122826, 180510, 260746, 370370, 518518, 715870, 976170, 1315470, 1753975, 2314936, 3027224, 3923845, 5044920, 6436200, 8152542, 10255896

Offset: 0

Vladeta Jovovic, Dec 30 1999

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-5,-8,3,19,4,-24,-15,15,24,-4,-19,-3,8,5,-3,-2,1).

Column k=3 of A138107.

Cf. A005993.

< 1/(1 - x^i), {i, 1, n^2 - n}], {x, 0, nn}], x] (* Geoffrey Critzer, Aug 07 2015 *)
CoefficientList[Series[(x^10 + 3 x^8 + 10 x^7 + 16 x^6 + 12 x^5 + 16 x^4 + 10 x^3 + 3 x^2 + 1)/((1 - x^3)^3 (1 - x^2)^4 (1 - x)^2), {x, 0, 33}], x] (* Vincenzo Librandi, Aug 08 2015 *)

Vec((1 + 3*x^2 + 10*x^3 + 16*x^4 + 12*x^5 + 16*x^6 + 10*x^7 + 3*x^8 + x^10)/((1 - x)^2*(1 - x^2)^4*(1 - x^3)^3) + O(x^40)) \\ Andrew Howroyd, Mar 16 2020

G.f.: (x^10+3*x^8+10*x^7+16*x^6+12*x^5+16*x^4+10*x^3+3*x^2+1) / ((1-x^3)^3*(1-x^2)^4*(1-x)^2).

A050929 Number of directed multigraphs with loops on 4 nodes with n arcs.

Original entry on oeis.org

1, 2, 11, 47, 198, 713, 2423, 7388, 21003, 55433, 137944, 324659, 729022, 1567139, 3242954, 6479759, 12547894, 23607614, 43267994, 77405064, 135435666, 232137202, 390371944, 644897542, 1047890293, 1676518363, 2643628813

Offset: 0

Vladeta Jovovic, Dec 30 1999

Robert Israel, Table of n, a(n) for n = 0..10000

Column k=4 of A138107.

Cf. A005993.

gf:= (x^26-x^25 + 4*x^24 + 18*x^23 + 63*x^22 + 151*x^21 + 402*x^20 + 790*x^19 + 1511*x^18 + 2353*x^17 + 3400*x^16 + 4296*x^15 + 5115*x^14 + 5266*x^13 + 5115*x^12 + 4296*x^11 + 3400*x^10 + 2353*x^9 + 1511*x^8 + 790*x^7 + 402*x^6 + 151*x^5 + 63*x^4 + 18*x^3 + 4*x^2-x + 1)/((x^4-1)^4*(x^3-1)^5*(x^2-1)^4*(x-1)^3):
S:= series(gf,x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Aug 07 2015

nn = 30; n = 4; CoefficientList[Series[CycleIndex[ Join[PairGroup[SymmetricGroup[n], Ordered], Permutations[Range[n*(n - 1) + 1, n*(n - 1) + n]], 2], s] /. Table[s[i] -> 1/(1 - x^i), {i, 1, n^2 - n}], {x, 0, nn}], x] (* Geoffrey Critzer, Aug 07 2015*)

G.f.: (x^26-x^25 + 4*x^24 + 18*x^23 + 63*x^22 + 151*x^21 + 402*x^20 + 790*x^19 + 1511*x^18 + 2353*x^17 + 3400*x^16 + 4296*x^15 + 5115*x^14 + 5266*x^13 + 5115*x^12 + 4296*x^11 + 3400*x^10 + 2353*x^9 + 1511*x^8 + 790*x^7 + 402*x^6 + 151*x^5 + 63*x^4 + 18*x^3 + 4*x^2-x + 1)/((x^4-1)^4*(x^3-1)^5*(x^2-1)^4*(x-1)^3).

A092353 Expansion of (1+x^3)/((1-x)^2*(1-x^3)^2).

Original entry on oeis.org

1, 2, 3, 7, 11, 15, 24, 33, 42, 58, 74, 90, 115, 140, 165, 201, 237, 273, 322, 371, 420, 484, 548, 612, 693, 774, 855, 955, 1055, 1155, 1276, 1397, 1518, 1662, 1806, 1950, 2119, 2288, 2457, 2653, 2849, 3045, 3270, 3495, 3720, 3976, 4232, 4488, 4777, 5066, 5355, 5679

Offset: 0

N. J. A. Sloane, Mar 20 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Cf. A005993.

seq(add(floor(i/3)^2, i=1..n+3), n=0..60); # Ridouane Oudra, Oct 19 2019

a[n_] := Sum[Floor[i/3]^2, {i,1,n+3}]; Table[a[n], {n, 0, 100}] (* Enrique Pérez Herrero, Mar 20 2012 *)

def A092353():
    a, b, c, m = 0, 0, 0, 0
    while True:
        yield (a*(a*(2*a+9)+13)+b*(b+1)*(2*b+1)+c*(c+1)*(2*c+1)+6)//6
        m = m + 1 if m < 2 else 0
        if   m == 0: a += 1
        elif m == 1: b += 1
        elif m == 2: c += 1
a = A092353()
print([next(a) for  in range(52)]) # _Peter Luschny, May 04 2016

G.f.: (1+x^3)/((1-x)^2*(1-x^3)^2) = (1+x^3)/((1-x)^4*(1+x+x^2)^2).
a(n) = Sum(i=1..n+3, floor(i/3)^2). - Enrique Pérez Herrero, Mar 20 2012
a(n) = (1/2)*(-4*t^3 + (2n-7)*t^2 + (4n-1)*t +2n +2), where t = floor(n/3). - Ridouane Oudra, Oct 19 2019

A139625 Table read by rows: T(n,k) is the number of strongly connected directed multigraphs with loops and no vertex of degree 0, with n arcs and k vertices, which are transitive (the existence of a path between two points implies the existence of an arc between those two points).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 6, 1, 10, 1, 19, 1, 28, 1, 1, 44, 2, 1, 60, 10, 1, 85, 31, 1, 110, 90, 1, 146, 222, 1, 182, 520, 1, 231, 1090, 1, 1, 280, 2180, 2, 1, 344, 4090, 11, 1

Offset: 1

Benoit Jubin, May 01 2008, Sep 01 2008

Length of the n^th row: floor(sqrt(n)).
These graphs are reflexive (each vertex has a self-loop), so T(n,k) = sum(A139621(n-k^2,m),m=0..k)
T(n,1) = 1, T(n,2) = A005993(n-4), T(n,3) = A050927(n-9), T(n,4) = A050929(n-16).
Row sums: A139630.

			Triangle begins:
  1
  1
  1
  1  1
  1  2
  1  6
  1 10
  1 19
  1 28  1

Cf. A139623, A139624.

A168281 Triangle T(n,m) = 2*(min(n - m + 1, m))^2 read by rows.

Original entry on oeis.org

2, 2, 2, 2, 8, 2, 2, 8, 8, 2, 2, 8, 18, 8, 2, 2, 8, 18, 18, 8, 2, 2, 8, 18, 32, 18, 8, 2, 2, 8, 18, 32, 32, 18, 8, 2, 2, 8, 18, 32, 50, 32, 18, 8, 2, 2, 8, 18, 32, 50, 50, 32, 18, 8, 2, 2, 8, 18, 32, 50, 72, 50, 32, 18, 8, 2, 2, 8, 18, 32, 50, 72, 72, 50, 32, 18, 8, 2, 2, 8, 18, 32, 50, 72, 98, 72

Offset: 1

Paul Curtz, Nov 22 2009

Row sums are A099956(n-1) = 2*A005993(n-1).
The flattened triangle is simply 2 followed by A137508.
If A106314 is interpreted as a triangle, T(n,m) = 2*A106314(n,m).

			The table starts in row n=1 with columns 1<=m<=n as:
  2;
  2,2;
  2,8,2;
  2,8,8,2;
  2,8,18,8,2;
  2,8,18,18,8,2;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..10011 (First 141 rows).

Cf. A005993, A099956, A137508, A106314.

A168281 := proc(n,m) 2*(min(n+1-m,m))^2 ; end proc:
seq(seq(A168281(n,m),m=1..n),n=1..20) ;

Table[Map[2 Min[n + # - 1, #]^2 &, Drop[#, -Boole@ EvenQ@ n] ~Join~ Reverse@ # &@ Range@ Floor[n/2]], {n, 2, 14}] // Flatten (* Michael De Vlieger, Jul 19 2016 *)

Rephrased all comments in terms of a triangle by R. J. Mathar, Nov 24 2010
More terms from Michael De Vlieger, Jul 19 2016
Definition corrected by Georg Fischer, Nov 11 2021

A225972 The number of binary pattern classes in the (2,n)-rectangular grid with 3 '1's and (2n-3) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Original entry on oeis.org

0, 0, 1, 6, 14, 32, 55, 94, 140, 208, 285, 390, 506, 656, 819, 1022, 1240, 1504, 1785, 2118, 2470, 2880, 3311, 3806, 4324, 4912, 5525, 6214, 6930, 7728, 8555, 9470, 10416, 11456, 12529, 13702, 14910, 16224, 17575, 19038, 20540, 22160, 23821, 25606, 27434, 29392

Offset: 0

Yosu Yurramendi, May 26 2013

Also the edge count of the n X n black bishop graph. - Eric W. Weisstein, Jun 26 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Black Bishop Graph
Eric Weisstein's World of Mathematics, Edge Count
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A226048, A000330.

Cf. A289179 (edge count of white bishop graph).

[(1/4)*(Binomial(2*(n-1),3)+2*Binomial(n-2,1)*(1/2)*(1+(-1)^n)): n in [1..50]]; // Vincenzo Librandi, Sep 04 2013

A225972:=n->(n-1)*(4*n^2-2*n-3*(-1)^n+3)/12; seq(A225972(n), n=0..40); # Wesley Ivan Hurt, Mar 02 2014

Table[(n - 1)*(4*n^2 - 2*n - 3*(-1)^n + 3)/12, {n, 0, 40}] (* Bruno Berselli, May 29 2013 *)
CoefficientList[Series[x^2 (1 + 4 x + x^2 + 2 x^3) / ((1 + x)^2 (1 - x)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 04 2013 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 1, 6, 14, 32, 55}, 20] (* Eric W. Weisstein, Jun 27 2017 *)

a <- vector()
    for(n in 0:40) a[n] <- (1/4)*(choose(2*(n-1),3) + 2*choose(n-2,1)*(1/2)*(1+(-1)^n))
    a  # Yosu Yurramendi and María Merino, Aug 21 2013

a(n) = A000330(n) + A142150(n) = (n-1)*(4*n^2 - 2*n - 3*(-1)^n + 3)/12.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) with n > 5, a(0)=0, a(1)=0, a(2)=1, a(3)=6, a(4)=14, a(5)=32.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 4*(n-4)*(-1)^n with n > 3, a(0)=0, a(1)=0, a(2)=1, a(3)=6.
G.f.: x^2*(1 + 4*x + x^2 + 2*x^3)/((1+x)^2*(1-x)^4). - Bruno Berselli, May 29 2013
a(n) = (1/4)*(binomial(2*(n-1),3) + 2*binomial(n-2,1)*(1/2)*(1+(-1)^n)). - Yosu Yurramendi and María Merino, Aug 21 2013
a(n) = A005993(n-2) + A199771(n-1), n >= 2. - Christopher Hunt Gribble, Mar 02 2014

More terms from Vincenzo Librandi, Sep 04 2013

cp's OEIS Frontend

A106314 Triangle T(n,k) composed of the squares min(n,k)^2.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A136564 Array read by rows: T(n,k) is the number of directed multigraphs with loops with n arcs, k vertices, and no vertex of degree 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A248011 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing three 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A248059 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing four 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A050927 Number of directed multigraphs with loops on 3 nodes with n arcs.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A050929 Number of directed multigraphs with loops on 4 nodes with n arcs.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A092353 Expansion of (1+x^3)/((1-x)^2*(1-x^3)^2).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A139625 Table read by rows: T(n,k) is the number of strongly connected directed multigraphs with loops and no vertex of degree 0, with n arcs and k vertices, which are transitive (the existence of a path between two points implies the existence of an arc between those two points).

Views

Author

Keywords

Comments

Examples

Crossrefs