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A000368 Number of connected graphs with one cycle of length 4.
(Formerly M3365 N1356)
5
1, 1, 4, 9, 28, 71, 202, 542, 1507, 4114, 11381, 31349, 86845, 240567, 668553, 1860361, 5188767, 14495502, 40572216, 113743293, 319405695, 898288484, 2530058013, 7135848125, 20152898513, 56986883801 (list; graph; refs; listen; history; text; internal format)
OFFSET
4,3
REFERENCES
F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, page 69.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 4..2000 (terms 4..43 from Sean A. Irvine, 44..200 from Washington Bomfim)
FORMULA
From Washington Bomfim, Jul 19 2012 and Dec 22 2020: (Start)
a(n) = Sum_{P}( g(Q) ), where P is the set of the partitions Q of n with 4 parts, Q with distinct parts D[1]..D[d], D[1] the part of maximum multiplicity m in Q, f(n) = A000081(n), and g(Q) given by,
| 3 * f(D[1]) * f(D[2]) * f(D[3]) * f(D[4]), if d = 4,
| (f(D[1])^4 + 2*f(D[1])^3 + 3*f(D[1])^2 + 2 * f(D[1])/8, if d = 1,
g(Q) = | f(D[1]) * f(D[2]) * f(D[3]) * (3 * f(D[1]) + 1)/2, if d = 3,
| ((3*f(D[2])^2+f(D[2]])*f(D[1])^2+(f(D[2])^2+3*f(D[2]])*f(D[1]])/4,
| if d=2, and m=2,
| f(D[1])^2 * f(D[2]) * (f(D[1]) + 1)/2, if d=2, and m=3.
(End)
G.f.: (2*t(x^4) + 3*t(x^2)^2 + 2*t(x)^2*t(x^2) + t(x)^4)/8 where t(x) is the g.f. of A000081. - Andrew Howroyd, Dec 03 2020
a(n) ~ (A187770 + A339986) * A051491^n / (2 * n^(3/2)). - Vaclav Kotesovec, Dec 25 2020
MATHEMATICA
Needs["Combinatorica`"]; nn = 30; s[n_, k_] := s[n, k] = a[n + 1 - k] + If[n < 2 k, 0, s[n - k, k]]; a[1] = 1; a[n_] := a[n] = Sum[a[i] s[n - 1, i] i, {i, 1, n - 1}]/(n - 1); rt = Table[a[i], {i, 1, nn}]; Take[CoefficientList[CycleIndex[DihedralGroup[4], s] /. Table[s[j] -> Table[Sum[rt[[i]] x^(k*i), {i, 1, nn}], {k, 1, nn}][[j]], {j, 1, nn}], x], {5, nn}] (* Geoffrey Critzer, Oct 12 2012, after code given by Robert A. Russell in A000081 *)
A000081 = Rest[Cases[ Import["https://oeis.org/A000081/b000081.txt", "Table"], {_, _}][[All, 2]]]; max = 30; g81 = Sum[A000081[[k]]*x^k, {k, 1, max}]; g81x2 = Sum[A000081[[k]]*x^(2 k), {k, 1, max}]; g81x4 = Sum[A000081[[k]]*x^(4 k), {k, 1, max}]; Drop[CoefficientList[ Series[(2*g81x4 + 3*g81x2^2 + 2*g81^2*g81x2 + g81^4)/8, {x, 0, max}], x], 4] (* Vaclav Kotesovec, Dec 25 2020 *)
PROG
(PARI) g(Q)={my(V=Vec(Q), D=Set(V), d=#D); if(d==4, return(3*f[D[1]]*f[D[2]]*f[D[3]]*f[D[4]]));
if(d==1, return((f[D[1]]^4+2*f[D[1]]^3+3*f[D[1]]^2+2*f[D[1]])/8));
my(k=1, m = #select(x->x == D[k], V), t); while(m==1, k++; m = #select(x->x == D[k], V)); t = D[1]; D[1] = D[k]; D[k] = t;
if(d == 3, return( f[D[1]] * f[D[2]] * f[D[3]] * (3 * f[D[1]] + 1)/2 ) );
if(m==3, return(f[D[1]]^2 * f[D[2]] * (f[D[1]] + 1)/2));
((3*f[D[2]]^2 + f[D[2]])*f[D[1]]^2 + (f[D[2]]^2 + 3*f[D[2]])*f[D[1]])/4 };
seq(max_n) = { my(s, a = vector(max_n), U); f = vector(max_n); f[1] = 1;
for(j=1, max_n - 1, if(j%100==0, print(j)); f[j+1] = 1/j * sum(k=1, j, sumdiv(k, d, d * f[d]) * f[j-k+1]));
for(n=4, max_n, s=0; forpart(Q = n, if( (Q[4] > Q[3]) && (Q[3]-1 > Q[2]),
U = U / (f[Q[4] + 1] * f[Q[3] - 1]) * f[Q[4]] * f[Q[3]], U = g(Q)); s += U,
[1, n], [4, 4]); a[n] = s; if(n % 100 == 0, print(n": " s))); a[4..max_n] };
\\ Washington Bomfim, Jul 19 2012 and Dec 22 2020
CROSSREFS
Column k=4 of A217781.
Second diagonal of A058879.
Sequence in context: A244968 A071258 A120333 * A232765 A094255 A192234
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Apr 20 2000
Definition improved by Franklin T. Adams-Watters, May 16 2006
More terms from Sean A. Irvine, Nov 14 2010
STATUS
approved

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Last modified March 28 04:13 EDT 2024. Contains 371235 sequences. (Running on oeis4.)