A035082 -id:A035082 - cp's OEIS frontend

A035086 Number of increasing rooted polygonal cacti (Husimi graphs) with n nodes.

1, 0, 1, 3, 19, 135, 1204, 12537, 150556, 2043930, 30969211, 517973148, 9478800604, 188381470095, 4040440921699, 93020386382742, 2287969523647171, 59877222907995675, 1661259526266784171, 48705364034046758493, 1504614657169716311674, 48848750173492332588525

Offset: 1

Christian G. Bower, Nov 15 1998

Nodes are numbered and the numbers increase as you move away from the root to any point on the same polygon.

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301 and Chapter 5.
F. Harary and E. M. Palmer, Graphical Enumeration, p. 71.

Alois P. Heinz, Table of n, a(n) for n = 1..200
F. Harary and R. Z. Norman, The Dissimilarity Characteristic of Husimi Trees, Annals of Mathematics, 58 1953, pp. 134-141.
F. Harary and G. E. Uhlenbeck, On the Number of Husimi Trees, Proc. Nat. Acad. Sci. USA vol. 39 pp. 315-322 1953.
Index entries for sequences related to cacti
Index entries for sequences related to rooted trees

Cf. A035082-A035088.

A:= proc(n) option remember; if n<=1 then x else convert(series(Int(exp(A(n-1)^2/ (2-2*A(n-1))), x), x=0, n+1), polynom) fi end; a:= n-> coeff(A(n), x, n)*n!: seq(a(n), n=1..22); # Alois P. Heinz, Aug 22 2008

max = 22; sy = Series[Integrate[E^(-(y^2/(2-2*y))), y], {y, 0, max}]; sx = Normal[ InverseSeries[sy, x]]; a[n_] := Coefficient[sx, x, n]*n!; Table[a[n], {n, 1, max }] (* Jean-François Alcover, Feb 24 2015 *)

E.g.f. satisfies A'(x) = exp(A(x)^2/(2-2*A(x))).

A035087 Number of labeled rooted polygonal cacti (Husimi graphs) with n nodes.

Original entry on oeis.org

1, 0, 3, 12, 135, 1440, 20895, 342720, 6585705, 142430400, 3449279295, 92207808000, 2699909867655, 85900402748160, 2951318065570875, 108894519775641600, 4294542443185019025, 180277244225580902400, 8025792422657714379675, 377695544010698833920000

Offset: 1

Christian G. Bower, Nov 15 1998

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301.
Harary and E. M. Palmer, Graphical Enumeration, p. 71
F. Harary and R. Z. Norman "The Dissimilarity Characteristic of Husimi Trees" Annals of Mathematics, 58 1953, pp. 134-141
F. Harary and G. E. Uhlenbeck "On the Number of Husimi Trees" Proc. Nat. Acad. Sci. USA vol. 39 pp. 315-322 1953

Cf. A035082-A035088.

A:= proc(n) option remember; if n<=1 then x else convert(series(x* exp(A(n-1)^2/ (2-2*A(n-1))), x=0, n+1), polynom) fi end: a:= n-> coeff(A(n), x, n)*n!: seq(a(n), n=1..30); # Alois P. Heinz, Aug 22 2008

Rest[CoefficientList[InverseSeries[Series[E^(x^2/(2*(x-1)))*x,{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

E.g.f. satisfies A(x)=x*exp(A(x)^2/(2-2*A(x))).

a(n) ~ (1-s)^2 * sqrt(2/(6-11*s+4*s^2)) * n^(n-1) / (s * exp(1 - s^2/(2*(1-s))))^n, where s = 0.5391888728108891165... is the root of the equation 2-4*s+s^3=0. - Vaclav Kotesovec, Jan 08 2014

A035350 "BIK" (reversible, indistinct, unlabeled) transform of A000237.

Original entry on oeis.org

1, 1, 2, 5, 15, 48, 164, 583, 2142, 8062, 30950, 120651, 476418, 1901311, 7656763, 31074151, 126963466, 521820340, 2155911512, 8948711597, 37299355151, 156054201936, 655134261795, 2758885471920, 11651193009013

Offset: 0

Christian G. Bower, Nov 15 1998

nonn

Christian G. Bower, Table of n, a(n) for n=0..500
C. G. Bower, Transforms (2)

Cf. A000083, A000237, A000314, A035082, A035349-A035357.

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

Christian G. Bower, Nov 15 1998

Cf. A000083, A000237, A000314, A035082, A035349-A035357.

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

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 *)

E.g.f. satisfies A(x) = x*exp((2*A(x)-A(x)^2)/(2-2*A(x))).

a(n) ~ (1-s)^2 * sqrt(2/((2-s)*(1+s-s^2))) * n^(n-1) / (s * exp((2+(s-4)*s)/(2-2*s)))^n, where s = 1/3*(4-2/(-17+3*sqrt(33))^(1/3) + (-17+3*sqrt(33))^(1/3)) = 0.456310987307923638429144... is the root of the equation 6*s - 4*s^2 + s^3 = 2. - Vaclav Kotesovec, Jan 08 2014

More terms from Alois P. Heinz, Aug 20 2008

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

Christian G. Bower, Nov 15 1998

Limit n->infinity (a(n)/n!)^(1/n) = 2.168573... - Vaclav Kotesovec, Feb 28 2014

Cf. A000083, A000237, A000314, A035082, A035349-A035357.

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

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 *)

E.g.f. satisfies A'(x) = exp((2*A(x)-A(x)^2)/(2-2*A(x))).

a(18) corrected by Alois P. Heinz, Aug 20 2008

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

Christian G. Bower, Nov 15 1998

nonn

C. G. Bower, Transforms (2)

Cf. A000083, A000237, A000314, A035082, A035349-A035357.

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

Christian G. Bower, Nov 15 1998

nonn

C. G. Bower, Transforms (2)

Cf. A000083, A000237, A000314, A035082, A035349-A035357.

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

Christian G. Bower, Nov 15 1998

nonn

Cf. A000083, A000237, A000314, A035082, A035349-A035357.

G.f.: A(x) = B(x) + C(x) - B(x)*D(x), where B, C, D respectively are g.f.s of A035353, A035354, A035355.

cp's OEIS Frontend

A035086 Number of increasing rooted polygonal cacti (Husimi graphs) with n nodes.

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A035087 Number of labeled rooted polygonal cacti (Husimi graphs) with n nodes.

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A035350 "BIK" (reversible, indistinct, unlabeled) transform of A000237.

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A035351 Number of labeled rooted polygonal cacti with bridges (mixed Husimi trees) with n nodes.

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A035352 Number of increasing rooted polygonal cacti with bridges (mixed Husimi trees) with n nodes.

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Author

Keywords

Comments

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Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A035354 "DHK" (bracelet, identity, unlabeled) transform of A035353.

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Keywords

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A035355 "BHK" (reversible, identity, unlabeled) transform of A035353.

Views

Author

Keywords

Links

Crossrefs

A035356 Number of asymmetric polygonal cacti with bridges (mixed Husimi trees).

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Author

Keywords

Links

Crossrefs

Formula