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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) Washington Bomfim, Illustration of initial terms 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. Cf. A000081, A000226, A001429, A005703. Second diagonal of A058879. Sequence in context: A244968 A071258 A120333 * A232765 A094255 A192234 Adjacent sequences: A000365 A000366 A000367 * A000369 A000370 A000371 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 25 19:21 EDT 2023. Contains 361528 sequences. (Running on oeis4.)