A290428 -id:A290428 - cp's OEIS frontend

A098568 Triangle of triangular binomial coefficients, read by rows, where column k has the g.f.: 1/(1-x)^((k+1)*(k+2)/2) for k >= 0.

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 21, 10, 1, 1, 15, 56, 55, 15, 1, 1, 21, 126, 220, 120, 21, 1, 1, 28, 252, 715, 680, 231, 28, 1, 1, 36, 462, 2002, 3060, 1771, 406, 36, 1, 1, 45, 792, 5005, 11628, 10626, 4060, 666, 45, 1, 1, 55, 1287, 11440, 38760, 53130, 31465, 8436

Offset: 0

Paul D. Hanna, Sep 15 2004

The row sums form A098569: {1,2,5,14,43,143,510,1936,7775,32869,...}. How do the terms of row k tend to be distributed as k grows?
Remarkably, column k of the matrix inverse (A121434) equals signed column k of the triangular matrix power: A107876^(k*(k+1)/2) for k >= 0. - Paul D. Hanna, Aug 25 2006
Surprisingly, the row sums (A098569) equal the row sums of triangle A131338. - Paul D. Hanna, Aug 30 2007
Number of sequences S = s(1)s(2)...s(n) such that S contains m 0's, for 1<=j<=n, s(j)Frank Ruskey, Apr 15 2011
As a rectangular array read by antidiagonals R(n,k) (n>=2, k>=0) is the number of labeled graphs on n nodes that have exactly k arcs where multiple arcs are allowed to connect distinct vertex pairs. R(n,k) = C(C(n,2)+k-1,k). See example below. - Geoffrey Critzer, Nov 12 2011

			G.f.s of columns: 1/(1-x), 1/(1-x)^3, 1/(1-x)^6, 1/(1-x)^10, 1/(1-x)^15, ...
Rows begin:
  1;
  1,  1;
  1,  3,    1;
  1,  6,    6,     1;
  1, 10,   21,    10,      1;
  1, 15,   56,    55,     15,      1;
  1, 21,  126,   220,    120,     21,      1;
  1, 28,  252,   715,    680,    231,     28,     1;
  1, 36,  462,  2002,   3060,   1771,    406,    36,     1;
  1, 45,  792,  5005,  11628,  10626,   4060,   666,    45,    1;
  1, 55, 1287, 11440,  38760,  53130,  31465,  8436,  1035,   55,  1;
  1, 66, 2002, 24310, 116280, 230230, 201376, 82251, 16215, 1540, 66, 1; ...
From _Frank Ruskey_, Apr 15 2011: (Start)
In reference to comment about s(1)s(2)...s(n) above,
   a(4,2) = 6 = |{0012, 0013, 0023, 0101, 0103, 0120}|  and
   a(4,3) = 6 = |{0001, 0002, 0003, 0010, 0020, 0100}|. (End)
From _Geoffrey Critzer_, Nov 12 2011: (Start)
In reference to comment about multigraphs above,
  1,    1,    1,    1,    1,     1,     ...  2 nodes
  1,    3,    6,    10,   15,    21,    ...  3 nodes
  1,    6,    21,   56,   126,   252,   ...  .
  1,    10,   55,   220,  715,   2002,  ...  .
  1,    15,   120,  680,  3060,  11628, ...  .
  1,    21,   231,  1771, 10626, 58130, ...  . (End)

Paul D. Hanna, Table of n, a(n) for n = 0..1080, of flattened triangle, read by rows 0..45.
Soheir M. Khamis, Height counting of unlabeled interval and N-free posets, Discrete Math. 275 (2004), no. 1-3, 165-175.
Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.
Zhicong Lin and Shishuo Fu, On 120-avoiding inversion and ascent sequences, arXiv:2003.11813 [math.CO], 2020.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO], (2017), table 60.

Cf. A098569. A290428 (unlabeled graphs).
Cf. A121434 (inverse); variants: A122175, A122176, A122177; A107876.
Cf. A131338.

t[n_, k_] = Binomial[(k+1)*(k+2)/2 + n-k-1, n-k]; Flatten[Table[t[n, k], {n, 0, 10}, {k, 0, n}]] (* Jean-François Alcover, Jul 18 2011 *)

{T(n,k)=binomial((k+1)*(k+2)/2+n-k-1,n-k)}
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))

T(n, k) = binomial((k+1)*(k+2)/2 + n-k-1, n-k).

A050531 Number of multigraphs with loops on 3 nodes with n edges.

Original entry on oeis.org

1, 2, 6, 14, 28, 52, 93, 152, 242, 370, 546, 784, 1103, 1512, 2040, 2706, 3534, 4554, 5803, 7304, 9108, 11252, 13780, 16744, 20205, 24206, 28826, 34126, 40176, 47056, 54857, 63648, 73542, 84630, 97014, 110808, 126139, 143108, 161868, 182546, 205282

Offset: 0

Vladeta Jovovic, Dec 29 1999

a(n) is also the number of multigraphs (no loops allowed) on 3 nodes with n edges of two colors. - Geoffrey Critzer, Aug 10 2015

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-3,0,6,0,-3,-2,1,2,-1).

Column k=3 of A290428.

Cf. A076118, A002620, A290428.

a076118:= gfun:-rectoproc({a(n+4) = 2*a(n+3)-3*a(n+2)+2*a(n+1)-a(n), a(0)=0,a(1)=1,a(2)=1,a(3)=-1}, a(n), remember):
f:= n -> ceil((-1)^n*a076118(n+1)/9+(-1)^n*n/32+(4009/4320)*n+(1/2)*n^2+(5/36)*n^3+(1/48)*n^4+(1/720)*n^5):
map(f, [$0..100]); # Robert Israel, Aug 07 2015

< 1/(1 - x^i), {i, 1, n^2 - n}], {x, 0, nn}], x] (* Geoffrey Critzer, Aug 07 2015 *)
CoefficientList[Series[(x^6 + x^4 + 2 x^3 + x^2 + 1)/((x^3 - 1)^2 (x^2 - 1)^2 (x - 1)^2), {x, 0, 45}], x] (* Vincenzo Librandi, Aug 08 2015 *)

Vec((x^6+x^4+2*x^3+x^2+1)/((x^3-1)^2*(x^2-1)^2*(x-1)^2) + O(x^40)) \\ Colin Barker, Jul 07 2019

G.f.: (x^6+x^4+2*x^3+x^2+1)/((x^3-1)^2*(x^2-1)^2*(x-1)^2).
a(n) = ceiling((-1)^n*A076118(n+1)/9+(-1)^n*n/32+(4009/4320)*n+(1/2)*n^2+(5/36)*n^3+(1/48)*n^4+(1/720)*n^5). - Robert Israel, Aug 07 2015
a(n) = (A+B+C)/6 where A = binomial(n+5,5); B = (n+2)*(n+3)*(n+4)/8 if n even, B = (n+1)*(n+3)*(n+5)/8 if n odd; C = 2*((n/3) + 1) if n divisible by 3, C = 0 if n not divisible by 3. - David Pasino, Jul 06 2019
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - 3*a(n-4) + 6*a(n-6) - 3*a(n-8) - 2*a(n-9) + a(n-10) + 2*a(n-11) - a(n-12) for n>11. - Colin Barker, Jul 07 2019

A050532 Number of multigraphs with loops on 4 nodes with n edges.

Original entry on oeis.org

1, 2, 7, 20, 53, 125, 287, 606, 1226, 2358, 4356, 7740, 13327, 22239, 36151, 57336, 88962, 135249, 201912, 296324, 428211, 609935, 857327, 1190216, 1633551, 2218011, 2981607, 3970548, 5241120, 6861024, 8911782, 11490282, 14711976, 18712911, 23653440

Offset: 0

Vladeta Jovovic, Dec 29 1999

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2,-2,-2,11,-1,1,-10,-10,12,4,12,-10,-10,1,-1,11,-2,-2,-2,-1,3,-1).

Column k=4 of A290428.

Cf. A002620.

<< Combinatorica`
nn = 30; n = 4; CoefficientList[Series[CycleIndex[Join[PairGroup[SymmetricGroup[n]], Permutations[Range[n*(n - 1)/2 + 1, n*(n + 1)/2]], 2], s] /.Table[s[i] -> 1/(1 - x^i), {i, 1, n^2 - n}], {x, 0, nn}], x] (* Geoffrey Critzer, Aug 07 2015 *)
CoefficientList[Series[(4 x^11 + 4 x^10 + 11 x^9 + 15 x^8 + 9 x^7 + 12 x^6 + 6 x^5 + 6 x^4 + 3 x^3 + 2 x^2 - x + 1)/((x^4 - 1)^2 (x^3 - 1)^3 (x^2 - 1)^2 (x-1)^3), {x, 0, 35}], x] (* Vincenzo Librandi, Aug 08 2015 *)

Vec((1 - x + 2*x^2 + 3*x^3 + 6*x^4 + 6*x^5 + 12*x^6 + 9*x^7 + 15*x^8 + 11*x^9 + 4*x^10 + 4*x^11)/((1 - x)^3*(1 - x^2)^2*(1 - x^3)^3*(1 - x^4)^2) + O(x^40)) \\ Andrew Howroyd, Mar 16 2020

G.f.: (4*x^11+4*x^10+11*x^9+15*x^8+9*x^7+12*x^6+6*x^5+6*x^4+3*x^3+2*x^2-x+1)/((x^4-1)^2*(x^3-1)^3*(x^2-1)^2*(x-1)^3).

a(33)-a(34) from Vincenzo Librandi, Aug 08 2015

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.

A327615 Irregular triangle read by rows: T(n,k) is the number of unlabeled multigraphs with loops allowed and n edges covering k vertices, n >= 1, 1 <= k <= 2*n.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 1, 5, 8, 6, 2, 1, 1, 8, 19, 25, 16, 7, 2, 1, 1, 11, 40, 73, 73, 47, 19, 7, 2, 1, 1, 15, 77, 194, 263, 232, 133, 58, 20, 7, 2, 1, 1, 19, 132, 454, 835, 951, 719, 397, 164, 61, 20, 7, 2, 1, 1, 24, 217, 984, 2385, 3507, 3365, 2306, 1177, 490, 175, 62, 20, 7, 2, 1

Offset: 1

Andrew Howroyd, Oct 23 2019

Covering k vertices means there are no vertices of degree zero.

			Triangle begins:
  1,  1;
  1,  3,   2,   1;
  1,  5,   8,   6,   2,   1;
  1,  8,  19,  25,  16,   7,   2,   1;
  1, 11,  40,  73,  73,  47,  19,   7,   2,  1;
  1, 15,  77, 194, 263, 232, 133,  58,  20,  7,  2, 1;
  1, 19, 132, 454, 835, 951, 719, 397, 164, 61, 20, 7, 2, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..650 (rows 1..25)

Row sums are A007717.
Columns k=2..3 are A024206, A327728.
Cf. A136564, A290428, A309936, A322115.

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)))}
C(n,m)={my(s=O(x*x^m)); forpart(p=n, s+=permcount(p)/edges(p, i->1-x^i+O(x*x^m))); Col(s/n!)}
T(m) = {my(n=2*m, A=Mat(vector(n+1, n, C(n-1,m)))); A[2..m+1, 2..n+1]-A[2..m+1, 1..n]}
{ my(A=T(8)); for(n=1, matsize(A)[1], print(A[n, 1..2*n])) }

T(n,k) = A290428(n,k) - A290428(n,k-1).

A360880 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) multigraphs with n edges and k nodes, loops allowed, n >= 1, 2 <= k <= n + 1.

Original entry on oeis.org

1, 2, 0, 4, 1, 0, 6, 2, 1, 0, 9, 6, 3, 1, 0, 12, 14, 13, 4, 1, 0, 16, 28, 39, 22, 5, 1, 0, 20, 52, 112, 98, 39, 6, 1, 0, 25, 93, 281, 383, 236, 63, 8, 1, 0, 30, 152, 655, 1304, 1220, 515, 102, 9, 1, 0, 36, 242, 1408, 3980, 5418, 3512, 1077, 153, 11, 1, 0

Offset: 1

Andrew Howroyd, Feb 25 2023

			Triangle T(n,k) begins (n edges >= 1, k vertices >= 2):
   1;
   2,   0;
   4,   1,   0;
   6,   2,   1,    0;
   9,   6,   3,    1,    0;
  12,  14,  13,    4,    1,   0;
  16,  28,  39,   22,    5,   1,   0;
  20,  52, 112,   98,   39,   6,   1, 0;
  25,  93, 281,  383,  236,  63,   8, 1, 0;
  30, 152, 655, 1304, 1220, 515, 102, 9, 1, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..190 (rows 1..19)

Columns k=2..3 are A002620(n+1), A050531(n-3).
Row sums are A360881.
Cf. A290428, A322115, A339070, A339160, A360870.

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A098568 Triangle of triangular binomial coefficients, read by rows, where column k has the g.f.: 1/(1-x)^((k+1)*(k+2)/2) for k >= 0.

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A050531 Number of multigraphs with loops on 3 nodes with n edges.

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A050532 Number of multigraphs with loops on 4 nodes with n edges.

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

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A327615 Irregular triangle read by rows: T(n,k) is the number of unlabeled multigraphs with loops allowed and n edges covering k vertices, n >= 1, 1 <= k <= 2*n.

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A360880 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) multigraphs with n edges and k nodes, loops allowed, n >= 1, 2 <= k <= n + 1.

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