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 A000368 Number of connected graphs with one cycle of length 4. (Formerly M3365 N1356) 4
 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 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). F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, page 69. LINKS Sean A. Irvine and Washington Bomfim, Table of n, a(n) for n = 4..200 (terms n=4..43 computed by Sean A. Irvine) Frank Ruskey, Combinatorial Generation Algorithm 4.26, p. 96 FORMULA a(n) = Sum_(P){ g(p,a) }, where P is the set of the partitions p of n with four parts, each p as a vector of nP distinct parts: p[1], p[2], ..., p[nP], with frequencies respectively a[1],a[2], ..., a[nP], a[1] = max(a[i], i = 1..nP). C(,) is a binomial coefficient, f = A000081, and g(p,a) is given below.         | 2*f(p[1]) + 3*C(f(p[1]),2)) * f(p[2]) * f(p[3])     if nP = 3,         | 3 * f(p[1]) * f(p[2]) * f(p[3]) * f(p[4])           if nP = 4,         | 3 * C(f(p[1]),2) * C(f(p[2]),2) + 2 * f(p[2]) * C(f(p[1]),2) + g(p,a)= < 2 * (f(p[1]) * (C(f(p[2]),2) + f(p[2]))) if nP=2, and a[1]= 2,         | f(p[2])*( f(p[1]) + 3*C(f(p[1]),3) + 2*f(p[1]) * (f(p[1])-1) )         |                                       if nP = 2, and a[1] = 3,         | f(p[1])*(2*f(p[1])-1+2*C(f(p[1])-1,2))+3*C(f(p[1]),4) if nP=1. -Washington Bomfim, Jul 19 2012 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 *) PROG (PARI) max_n = 200; f = vector(max_n);         \\ f[n] = A000081[n], n=1..max_n sum2(k) = {local(s); s=0; fordiv(k, d, s += d*f[d]); return(s)}; Init_f()={f[1]=1; for(n =1, max_n -2, s=0; for(k=1, n, s+=sum2(k)*f[n-k+1]); f[n+1]=s/n)}; S; P=[0, 1, 1, 1, 1, 0]; nP; a=[0, 0, 0, 0, 0]; p=[0, 0, 0, 0, 0]; C(x, y) = binomial(x, y); g()={if(nP==3, S+=(2*f[p[1]] + 3*C(f[p[1]], 2))*f[p[2]]*f[p[3]]; return()); if(nP == 4, S += 3 * f[p[1]] * f[p[2]] * f[p[3]] * f[p[4]]; return()); if((nP==2)&&(a[1]==2), S += 3*C(f[p[1]], 2)*C(f[p[2]], 2) + 2*f[p[2]]*C(f[p[1]], 2) + 2*(f[p[1]]*(C(f[p[2]], 2)+f[p[2]])); return()); if((nP==2)&&(a[1]==3), S += f[p[2]] * (f[p[1]] + 3*C(f[p[1]], 3) + 2*f[p[1]]*(f[p[1]]-1)); return()); S += f[p[1]] * ( 2*f[p[1]] - 1 + 2*C(f[p[1]]-1, 2) ) + 3*C(f[p[1]], 4); } Convert()={i=3; k=2; x=P[2]; nP=0;  \\ Convert partition P to (nP, p, a) while(1, while(P[i] == x, i++); nP++; p[nP] = P[k]; a[nP] = i-k;    if(a[nP] > a[1], b=a[1]; a[1]=a[nP]; a[nP]=b; b=p[1]; p[1]=p[nP]; p[nP]=b);    if(P[i] == 0, g(); return()); x = P[i]; k = i; i++) }                                         \\ F. Ruskey partition generator Part(n, k, s, t) = { P[t] = s; if((k == 1) || (n == k), Convert(), L = max(1, ceil((n - s)/(k - 1))); for(j = L, min(s, n-s-k+2), Part(n-s, k-1, j, t+1))); P[t] = 1; }; \\ A(n) = {S=0; Part(2*n, 4+1, n, 1); return(S)} Init_f(); for(n=4, max_n, print(n, " ", A(n)))          \\ b-file format \\ Washington Bomfim, Jul 19 2012 CROSSREFS Cf. A000081, A000226, A001429, A005703. 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 January 21 19:42 EST 2019. Contains 319350 sequences. (Running on oeis4.)