cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A333157 Triangle read by rows: T(n,k) is the number of n X n symmetric binary matrices with k ones in every row and column.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 10, 18, 10, 1, 1, 26, 112, 112, 26, 1, 1, 76, 820, 1760, 820, 76, 1, 1, 232, 6912, 35150, 35150, 6912, 232, 1, 1, 764, 66178, 848932, 1944530, 848932, 66178, 764, 1, 1, 2620, 708256, 24243520, 133948836, 133948836, 24243520, 708256, 2620, 1
Offset: 0

Views

Author

Andrew Howroyd, Mar 09 2020

Keywords

Comments

T(n,k) is the number of k-regular symmetric relations on n labeled nodes.
T(n,k) is the number of k-regular graphs with half-edges on n labeled vertices.
Terms may be computed without generating all graphs by enumerating the number of graphs by degree sequence. A PARI program showing this technique is given below. Burnside's lemma as applied in A122082 and A000666 can be used to extend this method to the case of unlabeled vertices A333159 and A333161 respectively.

Examples

			Triangle begins:
  1,
  1,   1;
  1,   2,     1;
  1,   4,     4,      1;
  1,  10,    18,     10,       1;
  1,  26,   112,    112,      26,      1;
  1,  76,   820,   1760,     820,     76,     1;
  1, 232,  6912,  35150,   35150,   6912,   232,   1;
  1, 764, 66178, 848932, 1944530, 848932, 66178, 764, 1;
  ...

Links

Crossrefs

Columns k=0..8 are A000012, A000085, A000986, A110040, A139670, A139671, A139673, A139674, A139675.
Row sums are A322698.
Central coefficients are A333164.
Cf. A188448 (transposed as array).
Cf. A059441, A295193, A333158, A333159, A333161.

Programs

  • PARI
    \\ See script in A295193 for comments.
GraphsByDegreeSeq(n, limit, ok)={
  local(M=Map(Mat([x^0,1])));
  my(acc(p,v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(r,p,i,q,v,e) = if(e<=limit && poldegree(q)<=limit, if(i<0, if(ok(x^e+q, r), acc(x^e+q, v)), my(t=polcoeff(p,i)); for(k=0,t,self()(r,p,i-1,(t-k+x*k)*x^i+q,binomial(t,k)*v,e+k)))));
  for(k=2, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], my(p=src[i,1]); recurse(n-k, p, poldegree(p), 0, src[i,2], 0))); Mat(M);
}
Row(n)={my(M=GraphsByDegreeSeq(n, n\2, (p,r)->poldegree(p)-valuation(p,x) <= r + 1), v=vector(n+1)); for(i=1, matsize(M)[1], my(p=M[i,1], d=poldegree(p)); v[1+d]+=M[i,2]; if(pollead(p)==n, v[2+d]+=M[i,2])); for(i=1, #v\2, v[#v+1-i]=v[i]); v}
for(n=0, 8, print(Row(n))) \\ Andrew Howroyd, Mar 14 2020

Formula

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

A139670 Number of n X n symmetric binary matrices with all row sums 4.

Original entry on oeis.org

1, 26, 820, 35150, 1944530, 133948836, 11234051976, 1127512146540, 133475706272700, 18406586045919060, 2925154024273348296, 530686776655470875076, 109004840145995702773410, 25164525076896596670014400, 6486836210471246515195539840, 1856264107759263993451053077856
Offset: 4

Views

Author

R. H. Hardin, Jun 12 2008

Keywords

Comments

From R. J. Mathar, Apr 07 2017: (Start)
These are the row sums of the following triangle, which shows in row n and column t the number of symmetric n X n {0,1}-matrices with trace t and 4 ones in each row and each column, 0 <= t <= n:
0: 1;
1: 0, 0;
2: 0, 0, 0;
3: 0, 0, 0, 0;
4: 0, 0, 0, 0, 1;
5: 1, 0, 10, 0, 15, 0;
6: 15, 0, 270, 0, 465, 0, 70;
7: 465, 0, 9660, 0, 19355, 0, 5670, 0;
(End)

Examples

			a(4) = 1:
  1 1 1 1
  1 1 1 1
  1 1 1 1
  1 1 1 1

Links

Crossrefs

Column k=4 of A333157 and row 4 of A188448.
Cf. A000085 (row sums 1), A000986 (row sums 2), A110040 (row sums 3).

Extensions

Terms a(13) and beyond from Andrew Howroyd, Mar 09 2020

A188448 T(n,k)=Number of (n*k)Xk binary arrays with nonzero rows in decreasing order, no more than 2 ones in any row and exactly n ones in every column.

Original entry on oeis.org

1, 2, 0, 4, 1, 0, 10, 4, 0, 0, 26, 18, 1, 0, 0, 76, 112, 10, 0, 0, 0, 232, 820, 112, 1, 0, 0, 0, 764, 6912, 1760, 26, 0, 0, 0, 0, 2620, 66178, 35150, 820, 1, 0, 0, 0, 0, 9496, 708256, 848932, 35150, 76, 0, 0, 0, 0, 0, 35696, 8372754, 24243520, 1944530, 6912, 1, 0, 0, 0, 0
Offset: 1

Views

Author

R. H. Hardin Mar 31 2011

Keywords

Comments

Table starts
.1.2.4.10..26...76...232.....764......2620........9496.........35696
.0.1.4.18.112..820..6912...66178....708256.....8372754.....108306280
.0.0.1.10.112.1760.35150..848932..24243520...805036704...30649435140
.0.0.0..1..26..820.35150.1944530.133948836.11234051976.1127512146540
.0.0.0..0...1...76..6912..848932.133948836.26615510712
.0.0.0..0...0....1...232...66178..24243520
.0.0.0..0...0....0.....1.....764
.0.0.0..0...0....0.....0
.0.0.0..0...0....0
.0.0.0..0...0

Examples

			All solutions for 9X3
..1..1..0
..1..0..1
..1..0..0
..0..1..1
..0..1..0
..0..0..1
..0..0..0
..0..0..0
..0..0..0

Links

Crossrefs

Row 1 is A000085
Row 2 is A000986
Row 3 is A110040
Row 4 is A139670
Row 5 is A139671
Row 6 is A139673
Row 7 is A139674
Row 8 is A139675

A284990 Triangle T(n,t) read by rows: the number of n X n {0,1} matrices with trace t where each row sum and each column sum is 3.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 6, 8, 9, 44, 210, 420, 610, 540, 216, 7570, 33120, 66870, 82080, 66870, 33120, 7570, 1975560, 8171730, 15729000, 18433415, 14372820, 7499940, 2398900, 357435, 749649145, 2971510080, 5508175260, 6267658544, 4815171270, 2570369760, 932429820, 209185200, 22040361
Offset: 0

Views

Author

R. J. Mathar, Apr 07 2017

Keywords

Examples

			0:        1
1:        0       0
2:        0       0        0
3:        0       0        0        1
4:        1       0        6        8        9
5:       44     210      420      610      540     216
6:     7570   33120    66870    82080    66870   33120    7570
7:  1975560 8171730 15729000 18433415 14372820 7499940 2398900 357435

Links

Crossrefs

Cf. A007107 (diagonal?), A001501 (row sums), A007105 (column 0?), A110040 (symmetric matrices).

Extensions

More terms from Alois P. Heinz, Apr 09 2017

A110041 a(n) = number of labeled graphs on n vertices (with no isolated vertices, multi-edges or loops) such that the degree of every vertex is at most 3.

Original entry on oeis.org

1, 0, 1, 4, 41, 512, 8285, 166582, 4054953, 116797432, 3912076929, 150190759240, 6532014077809, 318632936830136, 17286883399233149, 1035508343364348938, 68053563847088272945, 4879593083836366195728, 379847137967853770523937, 31960371880691511556886988
Offset: 0

Views

Author

Marni Mishna, Jul 08 2005

Keywords

Comments

P-recursive.

Examples

			Graphs listed by edgeset: a(3) = 4: {(1,2), (2,3)}, {(1,3), (2,3)}, {(1,3), (1,2)}, {(2,3), (1,2), (1,3)}.

References

  • Goulden, I. P.; Jackson, D. M. Labelled graphs with small vertex degrees and $P$-recursiveness. SIAM J. Algebraic Discrete Methods 7(1986), no. 1, 60--66. MR0819706 (87k:05093) [Gives e.g.f.]

Links

Crossrefs

Cf. A110040, A110039, A002829, A001205, A001147.

Formula

Satisfies the linear recurrence: (-150917976*n^2 - 105258076*n^3 - 1925*n^9 - 13339535*n^5 - 45995730*n^4 - 357423*n^7 - 2637558*n^6 - 120543840*n - n^11 - 66*n^10 - 39916800 - 32670*n^8)*a(n) + (22057180*n^4 + 2*n^10 + 69934280*n^3 + 140581872*n^2 + 161254080*n + 4621890*n^5 + 79833600 + 130*n^9 + 3720*n^8 + 61620*n^7 + 653226*n^6)*a(n + 1) +
(3*n^10 + 6932835*n^5 + 5580*n^8 + 92430*n^7 + 979839*n^6 + 241881120*n + 33085770*n^4 + 104901420*n^3 + 210872808*n^2 + 119750400 + 195*n^9)*a(n + 2) + (6932520*n^3 + 39916800 + 136080*n^5 + 24168936*n^2 + 9324*n^6 + 47363040*n + 1223334*n^4 + 6*n^8 + 360*n^7)*a(n + 3) + (6*n^8 + 1431654*n^4 + 372*n^7 + 9996*n^6 + 152040*n^5 + 59875200 + 8545908*n^3 + 31580424*n^2 + 66054960*n)*a(n + 4) + (9100956*n + 6*n^7 + 9646560 + 3631220*n^2 + 335*n^6 + 7929*n^5 + 103085*n^4 + 794709*n^3)*a(n + 5) +
(492*n^6 + 9*n^7 + 11032560 + 11359*n^5 + 143385*n^4 + 1067026*n^3 + 4671483*n^2 + 11110486*n)*a(n + 6) + (1021680 + 1041*n^4 + 17838*n^3 + 150699*n^2 + 626358*n + 24*n^5)*a(n + 7) + (461340 + 7027*n^3 + 9*n^5 + 61461*n^2 + 267044*n + 399*n^4)*a(n + 8) + (100980 + 5751*n^2 + 9*n^4 + 39408*n + 372*n^3)*a(n + 9) + (-6414*n - 588*n^2 - 18*n^3 - 23364)*a(n + 10) + (-48*n - 528)*a(n + 11) + 24*a(n + 12) = 0.
Differential equation satisfied by the exponential generating function: {F(0) = 1, 9*t^4*(t^4 + t + t^2 - 2)^2*(d^2/dt^2)F(t) + 3*t*(-4*t^6 + 8*t^5 - 16*t + t^10 - 16*t^2 + 2*t^7 + 8 - 2*t^4 + 2*t^8 + 10*t^3)*(t^4 + t + t^2 - 2)*(d/dt)F(t) - t^2*(t^4 + t + t^2 - 2)*(t^10 - 2*t^9 - 6*t^7 - 12*t^6 + t^5 - t^4 + 39*t^3 - 10*t^2 + 24)*F(t)}.
Satisfies the recurrence (of order 8): 12*(81*n^4 - 837*n^3 + 2997*n^2 - 4326*n + 1987)*a(n) = 18*(n-1)*(81*n^4 - 810*n^3 + 2709*n^2 - 3435*n + 1036)*a(n-1) + 3*(n-1)*(243*n^6 - 2997*n^5 + 14499*n^4 - 35118*n^3 + 44823*n^2 - 26766*n + 3244)*a(n-2) + 3*(n-2)*(n-1)*(81*n^5 - 1080*n^4 + 4968*n^3 - 9825*n^2 + 7666*n - 178)*a(n-3) + (n-3)*(n-2)*(n-1)*(243*n^5 - 2430*n^4 + 8721*n^3 - 13896*n^2 + 8637*n - 2468)*a(n-4) + (n-4)*(n-3)*(n-2)*(n-1)*(405*n^4 - 3537*n^3 + 11934*n^2 - 15915*n + 6008)*a(n-5) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(243*n^5 - 2916*n^4 + 11799*n^3 - 19593*n^2 + 11382*n + 502)*a(n-6) + (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(162*n^4 - 1026*n^3 + 2241*n^2 - 1884*n + 182)*a(n-7) - (n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(81*n^4 - 513*n^3 + 972*n^2 - 519*n - 98)*a(n-8). - Vaclav Kotesovec, Sep 10 2014
a(n) ~ 3^(n/2) * exp(sqrt(3*n) - 3*n/2 - 5/4) * n^(3*n/2) / 2^(n + 1/2) * (1 + 23/(24*sqrt(3*n))). - Vaclav Kotesovec, Nov 04 2023, extended Nov 06 2023
Limit_{n->oo} A110041(n)/A110040(n) = exp(2). - Vaclav Kotesovec, Nov 05 2023

Extensions

Edited and extended by Max Alekseyev, Apr 28 2010

A333163 Number of cubic graphs on n unlabeled nodes with half-edges.

Original entry on oeis.org

1, 0, 0, 1, 3, 4, 12, 24, 70, 172, 525, 1530, 5078, 16994, 61456, 228898, 895910, 3617148, 15130833, 65084088, 287828488, 1304327221, 6050218591, 28675928883, 138730847262, 684300453848, 3438439910436, 17585597712632, 91479580896616, 483699938173293, 2598090378779507
Offset: 0

Views

Author

Andrew Howroyd, Mar 11 2020

Keywords

Comments

A half-edge is like a loop except it only adds 1 to the degree of its vertex.
a(n) is the number of simple graphs on n unlabeled nodes with every node having degree 2 or 3.

Crossrefs

Column k=3 of A333161.
Cf. A005638, A110040, A186417.
Showing 1-6 of 6 results.

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