A381468
Number of simple connected graphs on n unlabeled nodes with no node a member of more than one cycle.
Original entry on oeis.org
1, 1, 1, 2, 4, 8, 20, 48, 133, 374, 1124, 3439, 10923, 35245, 116128, 387729, 1312038, 4485906, 15486546, 53900520, 188998450, 667062919, 2368440477, 8454560144, 30328595227, 109285433191, 395425965732, 1436219868659, 5234881134074, 19143123415166, 70216752517419
Offset: 0
A035351
Number of labeled rooted polygonal cacti with bridges (mixed Husimi trees) with n nodes.
Original entry on oeis.org
1, 2, 12, 124, 1810, 34056, 783874, 21331136, 669902076, 23845794400, 948733833256, 41721533664768, 2009539243299328, 105209055401980544, 5948937678563109000, 361296961279074942976, 23456120142707873968336, 1621073894248128387746304
Offset: 1
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A:= proc(n) option remember; if n<=0 then x else x* exp((2*A(n-1) -A(n-1)^2)/ (2-2*A(n-1))) fi end: a:= n-> coeff(series(A(n-1), x=0,n+1), x,n)*n!: seq(a(n), n=1..20); # Alois P. Heinz, Aug 20 2008
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Rest[CoefficientList[InverseSeries[Series[x/E^(((x-2)*x)/(2*(x-1))),{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 08 2014 *)
A381470
Number of simple connected graphs on n unlabeled nodes with exactly 2 non-overlapping cycles.
Original entry on oeis.org
1, 4, 21, 85, 345, 1289, 4713, 16622, 57535, 195212, 653318, 2158866, 7063333, 22906699, 73742762, 235863378, 750187968, 2374249283, 7481414941, 23482536967, 73449564533, 229016163367, 712044375528, 2208131225648, 6831543467752, 21089958138852, 64978894444220
Offset: 6
The a(6) = 1 graph is:
o o
/ \ / \
o---o---o---o
.
The a(7) = 4 graphs are:
o o---o o o o o---o o o o o
/ \ | | / \ / \ / \ / \ / \ / \ / / \
o---o---o---o o---o o---o o---o---o---o o---o---o---o
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\\ TreeGf gives gf of A000081.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={my(t=TreeGf(n), t2=subst(t,x,x^2), g=t*(t^2/(1-t) + t2*(1+t)/(1-t2))/2, g2=subst(g,x,x^2)); Vec(g^2/(1-t) + g2*(1+t)/(1-t2))/2}
A035352
Number of increasing rooted polygonal cacti with bridges (mixed Husimi trees) with n nodes.
Original entry on oeis.org
1, 1, 3, 16, 122, 1203, 14518, 207061, 3406083, 63465271, 1320938774, 30371545338, 764447981599, 20904838435264, 617151430504113, 19561785238965715, 662583041367287249, 23882958184429006800, 912777131398463190802, 36868849734952579404745
Offset: 1
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Ap:= proc(n) option remember; local A, f; if n<=0 then f:=1 else A:= Int(Ap(n-1),x); f:= exp((2*A -A^2)/ (2-2*A)) fi; convert(series(f, x, n+1), polynom) end: a:= n-> coeff(series(Ap(n-1), x=0,n), x,n-1)*(n-1)!: seq(a(n), n=1..30); # Alois P. Heinz, Aug 20 2008
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Ap[n_] := Ap[n] = Module[{A, f}, If[n <= 0, f=1, A = Integrate[Ap[n-1], x]; f = Exp[(2*A-A^2)/(2-2*A)]]; Series[f, {x, 0, n+1}] // Normal]; a[n_] := SeriesCoefficient[Ap[n-1], {x, 0, n-1}]*(n-1)!; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)
A035354
"DHK" (bracelet, identity, unlabeled) transform of A035353.
Original entry on oeis.org
1, 1, 1, 2, 4, 11, 33, 103, 330, 1077, 3556, 11895, 40187, 137049, 471070, 1630819, 5680854, 19899698, 70053093, 247711242, 879441789, 3133648932, 11202930544, 40172365842, 144453506494, 520759972064, 1881787195944, 6814766134276
Offset: 0
A035355
"BHK" (reversible, identity, unlabeled) transform of A035353.
Original entry on oeis.org
1, 1, 1, 2, 5, 15, 44, 140, 442, 1443, 4751, 15903, 53716, 183334, 630513, 2184417, 7614318, 26690344, 94015908, 332639104, 1181596784, 4212424485, 15066677338, 54050993664, 194438337125, 701224890628, 2534810109024
Offset: 0
A035356
Number of asymmetric polygonal cacti with bridges (mixed Husimi trees).
Original entry on oeis.org
1, 1, 0, 0, 0, 0, 1, 4, 13, 45, 143, 461, 1470, 4747, 15364, 50283, 165549, 549734, 1837539, 6184589, 20938983, 71297174, 244009225, 839101403, 2898145437, 10050549437, 34985611467, 122209803101, 428285696895, 1505491979826
Offset: 0
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