A005816 -id:A005816 - cp's OEIS frontend

A005815 Number of 4-valent labeled graphs with n nodes.

1, 0, 0, 0, 0, 1, 15, 465, 19355, 1024380, 66462606, 5188453830, 480413921130, 52113376310985, 6551246596501035, 945313907253606891, 155243722248524067795, 28797220460586826422720

Offset: 0

Simon Plouffe

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 411.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..260 (first 100 terms from Vincenzo Librandi)
Élie de Panafieu, Asymptotic expansion of regular and connected regular graphs, arXiv:2408.12459 [math.CO], 2024. See p. 9.
I. P. Goulden, D. M. Jackson, and J. W. Reilly, The Hammond series of a symmetric function and its application to P-recursiveness, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 179-193.
Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.
Vaclav Kotesovec, Recurrence (of order 10)
R. C. Read and N. C. Wormald, Number of labeled 4-regular graphs, J. Graph Theory, 4 (1980), 203-212.

Cf. A005814, A002829, A005816, A272905 (connected). A diagonal of A059441.

egf := (1+x-(1/3)*x^2-(1/6)*x^3)^(-1/2)*hypergeom([1/4, 3/4],[],-12*x*(x+2)*(x-1)/(x^3+2*x^2-6*x-6)^2)*exp(-x*(x^2-6)/(8*x+16));
ser := convert(series(egf,x=0,40),polynom):
seq(coeff(ser,x,i)*i!, i=0..degree(ser)); # Mark van Hoeij, Nov 07 2011

max = 17; f[x_] := HypergeometricPFQ[{1/4, 3/4}, {}, -12*x*(x + 2)*(x - 1)/(x^3 + 2*x^2 - 6*x - 6)^2]*Exp[-x*(x^2 - 6)/(8*x + 16)]/(1 + x - x^2/3 - x^3/6)^ (1/2); CoefficientList[Series[f[x], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Jun 19 2012, from e.g.f. *)

From Vladeta Jovovic, Mar 26 2001: (Start)
E.g.f. f(x) = Sum_{n >= 0} a(n)*x^n/(n)! satisfies the differential equation 16*x^2*(x - 1)^2*(x + 2)^2*(x^5 + 2*x^4 + 2*x^2 + 8*x - 4)*(d^2/dx^2)y(x) - 4*(x^13 + 4*x^12 - 16*x^10 - 10*x^9 - 36*x^8 - 220*x^7 - 348*x^6 - 48*x^5 + 200*x^4 - 336*x^3 - 240*x^2 + 416*x - 96)*(d/dx)y(x) - x^4*(x^5 + 2*x^4 + 2*x^2 + 8*x - 4)^2*y(x) = 0.
Recurrence: a(n) = - 1/384*(( - 256*n^2 - 896*n + 1152)*a(n - 1) + (768*n^3 - 3648*n^2 + 5568*n - 2688)*a(n - 2) + ( - 192*n^4 + 3264*n^3 - 14784*n^2 + 24384*n - 12672)*a(n - 3) + (224*n^6 - 4512*n^5 + 36304*n^4 - 148160*n^3 + 320016*n^2 - 341728*n + 137856)*a(n - 5) + ( - 640*n^5 + 8800*n^4 - 46400*n^3 + 116000*n^2 - 135360*n + 57600)*a(n - 4) + ( - 24*n^10 + 1320*n^9 - 31680*n^8 + 435600*n^7 - 3786552*n^6 + 21649320*n^5 - 82006320*n^4 + 201828000*n^3 - 306085824*n^2 + 255087360*n - 87091200)*a(n - 11) + (64*n^10 - 3480*n^9 + 82692*n^8 - 1127232*n^7 + 9726024*n^6 - 55255032*n^5 + 208179908*n^4 - 510068208*n^3 + 770738352*n^2 - 640484928*n + 218211840)*a(n - 9) + (16*n^11 - 992*n^10 + 27256*n^9 - 437160*n^8 + 4536288*n^7 - 31876656*n^6 + 154182488*n^5 - 510784360*n^4 + 1128552896*n^3 - 1570313952*n^2 + 1223830656*n - 397716480)*a(n - 10) + ( - 128*n^8 + 5488*n^7 - 94576*n^6 + 864976*n^5 - 4606672*n^4 + 14604352*n^3 - 26753984*n^2 + 25611264*n - 9630720)*a(n - 7) + (16*n^9 - 576*n^8 + 8704*n^7 - 71680*n^6 + 348880*n^5 - 1013824*n^4 + 1673376*n^3 - 1333120*n^2 + 226944*n + 161280)*a(n - 8) + (128*n^7 - 2192*n^6 + 12048*n^5 - 8240*n^4 - 151248*n^3 + 565312*n^2 - 765248*n + 349440)*a(n - 6) + ( - 4*n^13 + 364*n^12 - 14924*n^11 + 364364*n^10 - 5897892*n^9 + 66678612*n^8 - 540145892*n^7 + 3163772612*n^6 - 13344475144*n^5 + 39830815024*n^4 - 81255012384*n^3 + 106386868224*n^2 - 79211036160*n + 24908083200)*a(n - 14) + ( - 4*n^13 + 360*n^12 - 14612*n^11 + 353496*n^10 - 5674812*n^9 + 63680760*n^8 - 512439356*n^7 + 2983811688*n^6 - 12520194544*n^5 + 37201987680*n^4 - 75598952832*n^3 + 98660630016*n^2 - 73265264640*n + 22992076800)*a(n - 13) + ( - 16*n^12 + 1244*n^11 - 43208*n^10 + 884620*n^9 - 11860728*n^8 + 109396452*n^7 - 709293464*n^6 + 3243764260*n^5 - 10331326456*n^4 + 22203205904*n^3 - 30301280928*n^2 + 23300910720*n - 7504358400)*a(n - 12) + ( - n^14 + 105*n^13 - 5005*n^12 + 143325*n^11 - 2749747*n^10 + 37312275*n^9 - 368411615*n^8 + 2681453775*n^7 - 14409322928*n^6 + 56663366760*n^5 - 159721605680*n^4 + 310989260400*n^3 - 392156797824*n^2 + 283465647360*n - 87178291200)*a(n - 15)). (End)
a(n) = Sum_{d=0..floor(n/2), c=0..floor(n/2-d), b=0..(n-2c-2d), f=0..(n-2c-2d-b), k=0..min(n-b-2c-2d-f, 2n-2f-2b-3c-4d), j=0..floor(k/2+f)} ((-1)^(k+2f-j+d)*n!*(k+2f)!(2(2n-k-2f-2b-3c-4d))!) / (2^(5n-2k-2f-3b-8c-7d) * 3^(n-b-c-2d-k-f)*(2n-k-2f-2b-3c-4d)!*(k+2f-2j)!*j!*b!*c!*d!*k!*f!*(n-b-2c-2d-k-f)!). - Shanzhen Gao, Jun 05 2009
E.g.f.: (1+x-(1/3)*x^2-(1/6)*x^3)^(-1/2)*hypergeom([1/4, 3/4],[],-12*x*(x+2)*(x-1)/(x^3+2*x^2-6*x-6)^2)*exp(-x*(x^2-6)/(8*x+16)). - Mark van Hoeij, Nov 07 2011
a(n) ~ n^(2*n) * 2^(n+1/2) / (3^n * exp(2*n+15/4)). - Vaclav Kotesovec, Mar 11 2014

More terms from Vladeta Jovovic, Mar 26 2001

A333467 Array read by antidiagonals: T(n,k) is the number of k-regular multigraphs on n labeled nodes, loops allowed, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 2, 5, 3, 1, 1, 0, 3, 0, 17, 0, 1, 1, 1, 3, 15, 47, 73, 15, 1, 1, 0, 4, 0, 138, 0, 388, 0, 1, 1, 1, 4, 34, 306, 2021, 4720, 2461, 105, 1, 1, 0, 5, 0, 670, 0, 43581, 0, 18155, 0, 1, 1, 1, 5, 65, 1270, 25050, 291001, 1295493, 1256395, 152531, 945, 1

Offset: 0

Andrew Howroyd, Mar 23 2020

			Array begins:
=============================================================
n\k | 0   1     2       3        4          5           6
----+--------------------------------------------------------
  0 | 1   1     1       1        1          1           1 ...
  1 | 1   0     1       0        1          0           1 ...
  2 | 1   1     2       2        3          3           4 ...
  3 | 1   0     5       0       15          0          34 ...
  4 | 1   3    17      47      138        306         670 ...
  5 | 1   0    73       0     2021          0       25050 ...
  6 | 1  15   388    4720    43581     291001     1594340 ...
  7 | 1   0  2461       0  1295493          0   159207201 ...
  8 | 1 105 18155 1256395 50752145 1296334697 23544232991 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..405 (diagonals 0..27)

Rows n=0..3 are A000012, A059841, A008619, A006003.
Columns k=0..4 are A000012, A123023, A002135, A005814, A005816.
Cf. A059441 (graphs), A167625 (unlabeled nodes), A333351 (without loops).

b:= proc(l, i) option remember; (n-> `if`(n=0, 1,
     `if`(l[n]=0, b(sort(subsop(n=[][], l)), n-1),
     `if`(i<1, 0, b(l, i-1)+`if`(i=n, `if`(l[n]>1,
      b(subsop(n=l[n]-2, l), i), 0), `if`(l[i]>0,
      b(subsop(i=l[i]-1, n=l[n]-1, l), i), 0))))))(nops(l))
    end:
A:= (n, k)-> b([k$n], n):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 23 2020

b[l_, i_] := b[l, i] = Function[n, If[n == 0, 1, If[l[[n]] == 0, b[Sort[ ReplacePart[l, n -> Nothing]], n-1], If[i < 1, 0, b[l, i-1] + If[i == n, If[l[[n]] > 1, b[ReplacePart[l, n -> l[[n]]-2], i], 0], If[l[[i]] > 0, b[ReplacePart[l, {i -> l[[i]]-1, n -> l[[n]]-1}], i], 0]]]]]][Length[l]];
A[n_, k_] := b[Table[k, {n}], n];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 07 2020, after Alois P. Heinz *)

MultigraphsWLByDegreeSeq(n, limit, ok)={
  local(M=Map(Mat([0, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(r, h, p, q, v, e) = if(!p, if(ok(x^e+q, r), acc(x^e+q, v)), my(i=poldegree(p), t=pollead(p)); self()(r, limit, p-t*x^i, q+t*x^i, v, e); for(m=1, h-i, for(k=1, min(t, (limit-e)\m), self()(r, if(k==t, limit, i+m-1), p-k*x^i, q+k*x^(i+m), binomial(t, k)*v, e+k*m)))));
  for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], forstep(e=0, limit, 2, recurse(n-r, limit, src[i, 1], 0, src[i, 2], e)))); Mat(M);
}
T(n, k)={if(n%2&&k%2, 0, vecsum(MultigraphsWLByDegreeSeq(n, k, (p, r)->subst(deriv(p), x, 1)>=(n-2*r)*k)[, 2]))}
{ for(n=0, 8, for(k=0, 6, print1(T(n, k), ", ")); print) }

A188405 Number of (4*n) X n binary arrays with rows in nonincreasing order, 4 ones in every column and no more than 2 ones in any row.

Original entry on oeis.org

1, 1, 5, 42, 641, 14751, 478711, 20758650, 1158207312, 80758709676, 6877184737416, 701994697409136, 84574042067524470, 11870290445670605262, 1919446717950100963626, 354168049679464581788796, 73947210994621695613727526, 17342441149450781813176059990

Offset: 0

R. H. Hardin, Mar 30 2011

nonn

Number of n X n symmetric matrices with nonnegative integer entries and all row and column sums 4. - Andrew Howroyd, Apr 07 2020

In A005816 matrix elements on the diagonal are counted with a factor 2. This sequence here counts labeled multigraphs with n nodes (may be disconnected, undirected edges) without loops and degree at each node <=4. - R. J. Mathar, Jun 05 2022

			All solutions for 8X2
..1..1....1..1....1..1....1..0....1..1
..1..0....1..1....1..1....1..0....1..1
..1..0....1..1....1..0....1..0....1..1
..1..0....1..0....1..0....1..0....1..1
..0..1....0..1....0..1....0..1....0..0
..0..1....0..0....0..1....0..1....0..0
..0..1....0..0....0..0....0..1....0..0
..0..0....0..0....0..0....0..1....0..0

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

Row 4 of A188403.

Cf. A139670 (matrix elements 0 or 1).

a(0)=1 prepended and terms a(12) and beyond from Andrew Howroyd, Apr 07 2020

cp's OEIS Frontend

A005815 Number of 4-valent labeled graphs with n nodes.

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A333467 Array read by antidiagonals: T(n,k) is the number of k-regular multigraphs on n labeled nodes, loops allowed, n >= 0, k >= 0.

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A188405 Number of (4*n) X n binary arrays with rows in nonincreasing order, 4 ones in every column and no more than 2 ones in any row.

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