A035349 -id:A035349 - cp's OEIS frontend

A000083 Number of mixed Husimi trees with n nodes; or polygonal cacti with bridges.

1, 1, 1, 2, 4, 9, 23, 63, 188, 596, 1979, 6804, 24118, 87379, 322652, 1209808, 4596158, 17657037, 68497898, 268006183, 1056597059, 4193905901, 16748682185, 67258011248, 271452424286, 1100632738565, 4481533246014, 18319020658537, 75152228262785, 309337095953934

Offset: 0

N. J. A. Sloane

nonn

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).

Christian G. Bower, Table of n, a(n) for n = 0..500
G. W. Ford and G. E. Uhlenbeck, Combinatorial problems in the theory of graphs III, Proc. Nat. Acad. Sci. USA, 42 (1956), 529-535.
Eric Weisstein's World of Mathematics, Cactus Graph
Index entries for sequences related to cacti
Index entries for sequences related to trees

Cf. 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 A000237, A035349, A035350. - Christian G. Bower, Nov 15 1998

More terms from Christian G. Bower, Nov 15 1998

A000314 Number of mixed Husimi trees with n nodes; or labeled polygonal cacti with bridges.

Original entry on oeis.org

1, 1, 1, 4, 31, 362, 5676, 111982, 2666392, 74433564, 2384579440, 86248530296, 3476794472064, 154579941792256, 7514932528712896, 396595845237540600, 22581060079942183936, 1379771773100463174608, 90059660791562688208128, 6253914166368448348512064

Offset: 0

N. J. A. Sloane

nonn

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).

Alois P. Heinz, Table of n, a(n) for n = 0..100
G. W. Ford and G. E. Uhlenbeck, Combinatorial problems in the theory of graphs III, Proc. Nat. Acad. Sci. USA, 42 (1956), 529-535.
Index entries for sequences related to cacti
Index entries for sequences related to trees

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

A:= proc(n) option remember; if n<=0 then x else convert(series(x* exp((2*A(n-1) -A(n-1)^2)/ (2-2*A(n-1))),x=0,n+2), polynom) fi end: a:= n-> if n=0 then 1 else coeff(series(A(n-1), x=0,n+1), x,n)*(n-1)! fi: seq(a(n), n=0..30); # Alois P. Heinz, Aug 20 2008

A[n_] := A[n] = If[n <= 0, x, Normal[Series[x*Exp[(2*A[n-1]-A[n-1]^2)/ (2-2*A[n-1])], {x, 0, n+2}]]]; a[n_] := If[n == 0, 1, Coefficient [Series[A[n-1], {x, 0, n+1}], x, n]*(n-1)!]; Table [a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)

a(n) = A035351/n, n>0. - Christian G. Bower, Nov 15 1998

More terms from Christian G. Bower, Nov 15 1998

A000237 Number of mixed Husimi trees with n nodes; or rooted polygonal cacti with bridges.

Original entry on oeis.org

0, 1, 1, 3, 8, 26, 84, 297, 1066, 3976, 15093, 58426, 229189, 910127, 3649165, 14756491, 60103220, 246357081, 1015406251, 4205873378, 17497745509, 73084575666, 306352303774, 1288328048865, 5433980577776, 22982025183983

Offset: 0

N. J. A. Sloane

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).

Christian G. Bower, Table of n, a(n) for n=0..500
C. G. Bower, Transforms (2)
G. W. Ford and G. E. Uhlenbeck, Combinatorial problems in the theory of graphs III, Proc. Nat. Acad. Sci. USA, 42 (1956), 529-535.
N. J. A. Sloane, Transforms
Index entries for sequences related to cacti
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

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

BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[0]); for(n=1, n, v=concat([0,1], EulerT(Vec(BIK(Ser(v))-1)))); v} \\ Andrew Howroyd, Aug 30 2018

Shifts left under transform T where Ta = EULER(BIK(a)). [See Transforms links.] - Christian G. Bower, Nov 15 1998

More terms from Christian G. Bower, Nov 15 1998

A035357 Number of increasing asymmetric rooted polygonal cacti with bridges (mixed Husimi trees).

Original entry on oeis.org

1, 1, 1, 7, 39, 409, 4687, 62822, 945250, 15999616, 300150210, 6198330586, 139779046596, 3420083177362, 90241503643208, 2554721759776914, 77240614583288344, 2484170781778551036

Offset: 1

Christian G. Bower, Nov 15 1998

Cf. A000083, A000237, A000314, A035082, A035349-A035356.

Shifts left under transform T where Ta = EGJ(BHJ(a)).

A035353 Number of asymmetric rooted polygonal cacti with bridges (mixed Husimi trees).

Original entry on oeis.org

0, 1, 1, 1, 3, 7, 22, 67, 215, 692, 2283, 7599, 25631, 87211, 299386, 1035059, 3602083, 12606318, 44344764, 156698081, 555989604, 1980050697, 7075365521, 25360341963, 91155701023, 328500571740, 1186656421109, 4296084607302

Offset: 0

Christian G. Bower, Nov 15 1998

Andrew Howroyd, Table of n, a(n) for n = 0..200
C. G. Bower, Transforms (2)
N. J. A. Sloane, Transforms
Index entries for sequences related to cacti
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

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

BHK(p)={p + (1/(1-p) - (1+p)/subst(1-p, x, x^2))/2}
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=[0]); for(n=1, n, v=concat([0,1], WeighT(Vec(BHK(Ser(v)))))); v} \\ Andrew Howroyd, Aug 30 2018

Shifts left under transform T where Ta = WEIGH(BHK(a)).

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.

cp's OEIS Frontend

A000083 Number of mixed Husimi trees with n nodes; or polygonal cacti with bridges.

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A000314 Number of mixed Husimi trees with n nodes; or labeled polygonal cacti with bridges.

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Author

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References

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Programs

Maple

Mathematica

Formula

Extensions

A000237 Number of mixed Husimi trees with n nodes; or rooted polygonal cacti with bridges.

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Author

Keywords

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

A035357 Number of increasing asymmetric rooted polygonal cacti with bridges (mixed Husimi trees).

Views

Author

Keywords

Links

Crossrefs

Formula

A035353 Number of asymmetric rooted polygonal cacti with bridges (mixed Husimi trees).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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

Views