A138107 -id:A138107 - cp's OEIS frontend

A005993 Expansion of (1+x^2)/((1-x)^2*(1-x^2)^2).

1, 2, 6, 10, 19, 28, 44, 60, 85, 110, 146, 182, 231, 280, 344, 408, 489, 570, 670, 770, 891, 1012, 1156, 1300, 1469, 1638, 1834, 2030, 2255, 2480, 2736, 2992, 3281, 3570, 3894, 4218, 4579, 4940, 5340, 5740, 6181, 6622, 7106, 7590, 8119, 8648, 9224, 9800

Offset: 0

N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)

Alkane (or paraffin) numbers l(6,n).
Dimension of the space of homogeneous degree n polynomials in (x1, y1, x2, y2) invariant under permutation of variables x1<->y1, x2<->y2.
Also multidigraphs with loops on 2 nodes with n arcs (see A138107). - Vladeta Jovovic, Dec 27 1999
Euler transform of finite sequence [2,3,0,-1]. - Michael Somos, Mar 17 2004
a(n-2) is the number of plane partitions with trace 2. - Michael Somos, Mar 17 2004
With offset 4, a(n) is the number of bracelets with n beads, 3 of which are red, 1 of which is blue. For odd n, a(n) = C(n-1,3)/2. For even n, a(n) = C(n-1,3)/2 +(n-2)/4. For n >= 6, with K = (n-1)(n-2)/((n-5)(n-4)), for odd n, a(n) = K*a(n-2). For even n, a(n) = K*a(n-2) -(n-2)/(n-5). - Washington Bomfim, Aug 05 2008
Equals (1,2,3,4,...) convolved with (1,0,3,0,5,...). - Gary W. Adamson, Feb 16 2009
Equals row sums of triangle A177878.
Equals (1/2)*((1, 4, 10, 20, 35, 56, ...) + (1, 0, 2 0, 3, 0, 4, ...)).
From Ctibor O. Zizka, Nov 21 2014: (Start)
With offset 4, a(n) is the number of different patterns of the 2-color 4-partition of n.
P(n)_(k;t) gives the number of different patterns of the t-color, k-partition of n.
P(n)(k;t) = 1 + Sum(i=2..n) Sum(j=2..i) Sum(r=1..m) c(i,j)*v_r*F_r(X_1,...,X_i).
P(n;i;j) = Sum(r=1..m) c_(i,j)*v_r*F_r(X_1,...,X_i).
m partition number of i.
c_(i,j) number of different coloring patterns on the r-th form (X_1,...,X_i) of i-partition with j-colors.
v_r number of i-partitions of n of the r-th form (X_1,...,X_i).
F_r(X_1,...,X_i) number of different patterns of the r-th form i-partition of n.
Some simple results:
  P(1)(k;t)=1, P(2)(k;t)=2, P(3)(k;t)=4, P(4)(k;t)=11, etc.
  P(n;1;1) = P(n;n;n) = 1 for all n;
  P(n;2;2) = floor(n/2) (A004526);
  P(n;3;2) = (n*n - 2*n + n mod 2)/4 (A002620).
This sequence is a(n) = P(n;4;2).
2-coloring of 4-partition is (A,B,A,B) or (B,A,B,A).
Each 4-partition of n has one of the form (X_1,X_1,X_1,X_1),(X_1,X_1,X_1,X_2), (X_1,X_1,X_2,X_2),(X_1,X_1,X_2,X_3),(X_1,X_2,X_3,X_4).
The number of forms is m=5 which is the partition number of k=4.
Partition form (X_1,X_1,X_1,X_1) gives 1 pattern ((X_1A,X_1B,X_1A,X_1B), (X_1,X_1,X_1,X_2) gives 2 patterns, (X_1,X_1,X_2,X_2) gives 4 patterns, (X_1,X_1,X_2,X_3) gives 6 patterns and (X_1,X_2,X_3,X_4) gives 12 patterns.
Thus a(n) = P(n;4;2) = 1*1*v_1 + 1*2*v_2 + 1*4*v_3 + 1*6*v_4 + 1*12*v_5 where v_r is the number of different 4-partitions of the r-th form (X_1,X_2,X_3,X_4) for a given n.
Example:
The 4-partitions of 8 are (2,2,2,2), (1,1,1,5), (1,1,3,3), (1,1,2,4), and (1,2,2,3):
  (2,2,2,2) 1 pattern
  (1,1,1,5), (1,1,5,1) 2 patterns
  (1,1,3,3), (1,3,3,1), (3,1,1,3), (1,3,1,3) 4 patterns
  (1,1,2,4), (1,1,4,2), (1,2,1,4), (1,2,4,1), (1,4,1,2), (2,1,1,4) 6 patterns
  (2,2,1,3), (2,2,3,1), (2,1,2,3), (2,1,3,2), (2,3,2,1), (1,2,2,3) 6 patterns
Thus a(8) = P(8,4,2) = 1 + 2 + 4 + 6 + 6 = 19. (End)
a(n) = length of run n+2 of consecutive 1's in A254338. - Reinhard Zumkeller, Feb 27 2015
Take a chessboard of (n+2) X (n+2) unit squares in which the a1 square is black. a(n) is the number of composite squares having black unit squares on their vertices. - Ivan N. Ianakiev, Jul 19 2018
a(n) is the number of 1423-avoiding odd Grassmannian permutations of size n+2. Avoiding any of the patterns 2314 or 3412 gives the same sequence. - Juan B. Gil, Mar 09 2023

			a(2) = 6, since ( x1*y1, x2*y2, x1*x1+y1*y1, x2*x2+y2*y2, x1*x2+y1*y2, x1*y2+x2*y1 ) are a basis for homogeneous quadratic invariant polynomials.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
L. Smith, Polynomial Invariants of Finite Groups, A K Peters, 1995, p. 96.

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5, Lemma 6 3rd line.
Washington Bomfim, The 19 bracelets with 8 beads - one blue, three reds and four blacks. [From _Washington Bomfim_, Aug 05 2008]
T. M. Brown, On the unimodality of convolutions of sequences of binomial coefficients, arXiv:1810.08235 [math.CO] (2018).
Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017.
Dragomir Z. Djokovic, Poincaré series of some pure and mixed trace algebras of two generic matrices, arXiv:math/0609262 [math.AC], 2006. See Table 8.
Juan B. Gil and Jessica A. Tomasko, Pattern-avoiding even and odd Grassmannian permutations, arXiv:2207.12617 [math.CO], 2022.
Naihuan Jing, Kailash Misra, and Carla Savage, On multi-color partitions and the generalized Rogers-Ramanujan identities, arXiv:math/9907183 [math.CO], 1999.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
N. J. A. Sloane, Classic Sequences
L. Smith, Polynomial invariants of finite groups. A survey of recent developments. Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 3, 211-250. See page 218. MR1433171 (98i:13009).
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A177878.
Partial sums of A008794 (without 0). - Bruno Berselli, Aug 30 2013
Cf. A254338, A002260, A005408, A282011.
Cf. A008619, A005995, A018211, A018213, A062136.

Following Gary W. Adamson.
import Data.List (inits, intersperse)
a005993 n = a005994_list !! n
a005993_list = map (sum . zipWith (*) (intersperse 0 [1, 3 ..]) . reverse) $
                   tail $ inits [1..]
-- Reinhard Zumkeller, Feb 27 2015

I:=[1,2,6,10,19,28]; [n le 6 select I[n] else 2*Self(n-1)+Self(n-2)-4*Self(n-3)+Self(n-4)+2*Self(n-5)-Self(n-6): n in [1..60]]; // Vincenzo Librandi, Jul 19 2015

g := proc(n) local i; add(floor(i/2)^2,i=1..n+1) end: # Joseph S. Riel (joer(AT)k-online.com), Mar 22 2002
a:= n-> (Matrix([[1, 0$3, -1, -2]]).Matrix(6, (i,j)-> if (i=j-1) then 1 elif j=1 then [2, 1, -4, 1, 2, -1][i] else 0 fi)^n)[1,1]; seq (a(n), n=0..44); # Alois P. Heinz, Jul 31 2008

CoefficientList[Series[(1+x^2)/((1-x)^2*(1-x^2)^2),{x,0,44}],x]  (* Jean-François Alcover, Apr 08 2011 *)
LinearRecurrence[{2,1,-4,1,2,-1},{1,2,6,10,19,28},50] (* Harvey P. Dale, Feb 20 2012 *)

a(n)=polcoeff((1+x^2)/(1-x)^2/(1-x^2)^2+x*O(x^n),n)

a(n) = (binomial(n+3, n) + (1-n%2)*binomial((n+2)/2, n>>1))/2 \\ Washington Bomfim, Aug 05 2008

a = vector(50); a[1]=1; a[2]=2;
for(n=3, 50, a[n] = ((n+2)*a[n-2]+2*a[n-1]-n)/(n-2)); a \\ Gerry Martens, Jun 03 2018

def A005993():
    a, b, to_be = 0, 0, True
    while True:
        yield (a*(a*(2*a+9)+13)+b*(b+1)*(2*b+1)+6)//6
        if to_be: b += 1
        else: a += 1
        to_be = not to_be
a = A005993()
[next(a) for  in range(48)] # _Peter Luschny, May 04 2016

l(c, r) = 1/2 C(c+r-3, r) + 1/2 d(c, r), where d(c, r) is C((c + r - 3)/2, r/2) if c is odd and r is even, 0 if c is even and r is odd, C((c + r - 4)/2, r/2) if c is even and r is even, C((c + r - 4)/2, (r - 1)/2) if c is odd and r is odd.
G.f.: (1+x^2)/((1-x)^2*(1-x^2)^2) = (1+x^2)/((1+x)^2*(x-1)^4) = (1/(1-x)^4 +1/(1-x^2)^2)/2.
a(2n) = (n+1)(2n^2+4n+3)/3, a(2n+1) = (n+1)(n+2)(2n+3)/3. a(-4-n) = -a(n).
From Yosu Yurramendi, Sep 12 2008: (Start)
a(n+1) = a(n) + A008794(n+3) with a(1)=1.
a(n) = A027656(n) + 2*A006918(n).
a(n+2) = a(n) + A000982(n+2) with a(1)=1, a(2)=2. (End)
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6). - Jaume Oliver Lafont, Dec 05 2008
a(n) = (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
a(n) = (1/12)*n*(n+1)*(n+2) + (1/4)*(n+1)*(1/2)*(1-(-1)^n), with offset 1. - Yosu Yurramendi, Jun 20 2013
a(n) = Sum_{i=0..n+1} ceiling(i/2) * round(i/2) = Sum_{i=0..n+2} floor(i/2)^2. - Bruno Berselli, Aug 30 2013
a(n) = (n + 2)*(3*(-1)^n + 2*n^2 + 8*n + 9)/24. - Ilya Gutkovskiy, May 04 2016
Recurrence formula:  a(n) = ((n+2)*a(n-2)+2*a(n-1)-n)/(n-2), a(1)=1, a(2)=2. - Gerry Martens, Jun 10 2018
E.g.f.: exp(-x)*(6 - 3*x + exp(2*x)*(18 + 39*x + 18*x^2 + 2*x^3))/24. - Stefano Spezia, Feb 23 2020
a(n) = Sum_{j=0..n/2} binomial(c+2*j-1,2*j)*binomial(c+n-2*j-1,n-2*j) where c=2. For other values of c we have: A008619 (c=1), A005995 (c=3), A018211 (c=4), A018213 (c=5), A062136 (c=6). - Miquel A. Fiol, Sep 24 2024

A052171 Number of directed multigraphs with loops on an infinite set of nodes containing a total of n arcs.

Original entry on oeis.org

1, 2, 11, 52, 296, 1724, 11060, 74527, 533046, 3999187, 31412182, 257150093, 2188063401, 19299062896, 176059781439, 1657961491087, 16089088019098, 160643776819423, 1648068916722737, 17351137043998280, 187255329043638437, 2069426416836401375, 23397468305569068113, 270406562951254606048, 3191908298072118225550, 38454691427657997701136

Offset: 0

Vladeta Jovovic, Jan 26 2000

nonn

Row sums of A136564, limiting values of A138107. - Benoit Jubin, May 13 2008

Euler transform of A137975. - M. F. Hasler, Jul 31 2017

Andrew Howroyd, Table of n, a(n) for n = 0..50
Banglei Guan, Ji Zhao, and Laurent Kneip, Six-Point Method for Multi-Camera Systems with Reduced Solution Space, arXiv:2402.18066 [cs.CV], 2024. See p. 10.
J.-C. Novelli, J.-Y. Thibon and N. M. Thiery, Algèbres de Hopf de graphes, C.R. Acad. Sci. Paris (Comptes Rendus Mathématique), 339 (2004), 607-610.
Sanjaye Ramgoolam, Permutation Invariant Gaussian Matrix Models, arXiv:1809.07559 [hep-th], 2018.

Cf. A050535, A007717, A005993, A050927, A050929.

Cf. A104209. Cf. A137975 (connected).

a(n) = A138107(2*n,n). - Max Alekseyev, Oct 17 2017

a(16)-a(25) from Max Alekseyev, Jun 21 2011

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

A290428 Array read by antidiagonals: T(n,k) is the number of graphs with n edges and k vertices, allowing loops and multi-edges.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 4, 1, 0, 1, 2, 6, 6, 1, 0, 1, 2, 7, 14, 9, 1, 0, 1, 2, 7, 20, 28, 12, 1, 0, 1, 2, 7, 22, 53, 52, 16, 1, 0, 1, 2, 7, 23, 69, 125, 93, 20, 1, 0, 1, 2, 7, 23, 76, 198, 287, 152, 25, 1, 0, 1, 2, 7, 23, 78, 245, 550, 606, 242, 30, 1, 0

Offset: 0

R. J. Mathar, Jul 31 2017

Variant of A138107, here for non-directed edges.

			1   1   1   1    1    1    1   1   1...
0   1   2   2    2    2    2   2   2...
0   1   4   6    7    7    7   7   7...
0   1   6  14   20   22   23  23  23...
0   1   9  28   53   69   76  78  79...
0   1  12  52  125  198  245 264 271...
0   1  16  93  287  550  782 915 973...
0   1  20 152  606 1441 2392
0   1  25 242 1226 3611

Andrew Howroyd, Table of n, a(n) for n = 0..1325
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO], 2017. See Table 59.

Cf. A050531 (column 3), A050532 (column 4), A138107, A098568 (vertex-labeled).

rows = 12;
permcount[v_] := Module[{m=1, s=0, k=0, t}, For[i=1, i <= Length[v], i++, t = v[[i]]; k = If[i>1 && t == v[[i-1]], k+1, 1]; m *= t*k; s += t]; s!/m];
edges[v_, t_] := Product[g = GCD[v[[i]], v[[j]] ]; t[v[[i]]*v[[j]]/g]^g, {i, 2, Length[v]}, {j, 1, i - 1}]*Product[c = v[[i]]; t[c]^Quotient[c + 1, 2]*If[OddQ[c], 1, t[c/2]], {i, 1, Length[v]}];
col[k_] := col[k] = Module[{s = O[x]^rows}, Do[s += permcount[p]*1/edges[p, 1 - x^# + O[x]^rows&], {p, IntegerPartitions[k]}]; s/k!] // CoefficientList[#, x]&;
T[0, ] = 1; T[, 0] = 0;
T[n_, k_] := col[k][[n + 1]];
Table[T[n-k, k], {n, 0, rows-1}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jan 08 2021, after Andrew Howroyd *)

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v,t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i],v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c+1)\2)*if(c%2, 1, t(c/2)))}
T(m, n=m) = {Mat(vector(n+1, n, my(s=O(x*x^m)); forpart(p=n-1, s+=permcount(p)*1/edges(p,i->1-x^i+O(x*x^m))); Col(s/(n-1)!)))}
{ my(A=T(8)); for(n=1, #A, print(A[n,])) } \\ Andrew Howroyd, Oct 22 2019

A214398 Triangle where the g.f. of column k is 1/(1-x)^(k^2) for k>=1, as read by rows n>=1.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 10, 9, 1, 1, 20, 45, 16, 1, 1, 35, 165, 136, 25, 1, 1, 56, 495, 816, 325, 36, 1, 1, 84, 1287, 3876, 2925, 666, 49, 1, 1, 120, 3003, 15504, 20475, 8436, 1225, 64, 1, 1, 165, 6435, 54264, 118755, 82251, 20825, 2080, 81, 1, 1, 220, 12870, 170544

Offset: 1

Paul D. Hanna, Jul 15 2012

This is also the array A(n,k) read upwards antidiagonals, where the entry in row n and column k counts the vertex-labeled digraphs with n arcs and k vertices, allowing multi-edges and multi-loops (labeled analog to A138107). The binomial formula counts the weak compositions of distributing n arcs over the k^2 positions in the adjacency matrix. - R. J. Mathar, Aug 03 2017

			Triangle begins:
1;
1, 1;
1, 4, 1;
1, 10, 9, 1;
1, 20, 45, 16, 1;
1, 35, 165, 136, 25, 1;
1, 56, 495, 816, 325, 36, 1;
1, 84, 1287, 3876, 2925, 666, 49, 1;
1, 120, 3003, 15504, 20475, 8436, 1225, 64, 1;
1, 165, 6435, 54264, 118755, 82251, 20825, 2080, 81, 1;
1, 220, 12870, 170544, 593775, 658008, 270725, 45760, 3321, 100, 1; ...

Paul D. Hanna, Rows n = 0..45, flattened.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 (2017) Table 80.

Cf. A214400 (central terms), A178325 (row sums), A054688, A000290 (1st subdiagonal), A037270 (2nd subdiagonal).

Cf. A230049.

A214398 := proc(n,k)
    binomial(k^2+n-k-1,n-k) ;
end proc:
seq(seq(A214398(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Aug 03 2017

nmax = 11;
T[n_, k_] := SeriesCoefficient[1/(1-x)^(k^2), {x, 0, n-k}];
Table[T[n, k], {n, 1, nmax}, {k, 1, n}] // Flatten

T(n,k)=binomial(k^2+n-k-1,n-k)
for(n=1,11,for(k=1,n,print1(T(n,k),", "));print(""))

T(n,k) = binomial(k^2+n-k-1, n-k).
Row sums form A178325.
Central terms form A214400.
T(n,n-2) = A037270(n-2). - R. J. Mathar, Aug 03 2017
T(n,n-3) = (n^2-6*n+11)*(n^2-6*n+10)*(n-3)^2 /6. - R. J. Mathar, Aug 03 2017

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).

A333361 Array read by antidiagonals: T(n,k) is the number of directed loopless multigraphs with n arcs and k vertices.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 1, 1, 5, 2, 0, 0, 1, 1, 6, 10, 3, 0, 0, 1, 1, 6, 20, 24, 3, 0, 0, 1, 1, 6, 23, 69, 42, 4, 0, 0, 1, 1, 6, 24, 110, 196, 83, 4, 0, 0, 1, 1, 6, 24, 126, 427, 554, 132, 5, 0, 0, 1, 1, 6, 24, 129, 603, 1681, 1368, 222, 5, 0, 0

Offset: 0

Andrew Howroyd, Mar 16 2020

			==================================================
n\k | 0 1 2   3    4     5     6      7      8
----+---------------------------------------------
  0 | 1 1 1   1    1     1     1      1      1 ...
  1 | 0 0 1   1    1     1     1      1      1 ...
  2 | 0 0 2   5    6     6     6      6      6 ...
  3 | 0 0 2  10   20    23    24     24     24 ...
  4 | 0 0 3  24   69   110   126    129    130 ...
  5 | 0 0 3  42  196   427   603    668    684 ...
  6 | 0 0 4  83  554  1681  2983   3811   4116 ...
  7 | 0 0 4 132 1368  5881 13681  20935  24979 ...
  8 | 0 0 5 222 3240 19448 59680 112943 154504 ...
  ...

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

Columns k=0..4 are A000007, A000007, A008619, A037240, A050930.

Cf. A052170, A138107, A192517, A290428.

\\ here G(k,x) gives column k as rational function.
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^(2*g))) * prod(i=1, #v, t(v[i])^(v[i]-1))}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)/edges(p, i->1-x^i)); s/n!}
T(n)={Mat(vector(n+1, k, Col(O(y*y^n) + G(k-1, y + O(y*y^n)))))}
{my(A=T(10)); for(n=1, #A, print(A[n, ]))}

T(n,k) = A052170(n) for k >= 2*n.

cp's OEIS Frontend

A005993 Expansion of (1+x^2)/((1-x)^2*(1-x^2)^2).

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A052171 Number of directed multigraphs with loops on an infinite set of nodes containing a total of n arcs.

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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.

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A290428 Array read by antidiagonals: T(n,k) is the number of graphs with n edges and k vertices, allowing loops and multi-edges.

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A214398 Triangle where the g.f. of column k is 1/(1-x)^(k^2) for k>=1, as read by rows n>=1.

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

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

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