cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 18 results. Next

A035083 DIK(b)-DIK[ 2 ](b)-b where b is A035082.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 7, 14, 33, 74, 180, 438, 1090, 2741, 6994, 17966, 46565, 121440, 318597, 839953, 2224486, 5914248, 15780662, 42241422, 113402369, 305254039, 823690961, 2227640597, 6037142355, 16392945284, 44592703836
Offset: 0

Views

Author

Christian G. Bower, Nov 15 1998

Keywords

Links

Crossrefs

Cf. A035082, A035084, A035085.

Programs

  • PARI
    BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}
DIK(p,n)={(sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d))) + ((1+p)^2/(1-subst(p, x, x^2))-1)/2)/2}
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(p=O(x)); for(n=1, n, p=x+x^2*Ser(EulerT(Vec(BIK(p)-1)-Vec(p)))); Vec(DIK(p, n) - p - (p^2 + subst(p, x, x^2))/2, -(n+1))} \\ Andrew Howroyd, Aug 31 2018

A035084 BIK(b)-b where b is A035082.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 9, 20, 47, 112, 273, 676, 1694, 4296, 10991, 28350, 73614, 192327, 505093, 1332801, 3531598, 9393501, 25070735, 67121670, 180216260, 485133376, 1309101329, 3540394176, 9594562328, 26051397890, 70861839620
Offset: 0

Views

Author

Christian G. Bower, Nov 15 1998

Keywords

Links

Crossrefs

Cf. A035082, A035083, A035085.

Programs

  • PARI
    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(p=O(x)); for(n=1, n, p=x+x^2*Ser(EulerT(Vec(BIK(p)-1)-Vec(p)))); concat([0], Vec(BIK(p)-1)-Vec(p))} \\ Andrew Howroyd, Aug 30 2018

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

References

  • 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

Crossrefs

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

Formula

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

Extensions

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

Views

Author

N. J. A. Sloane

Keywords

References

  • 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

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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

Extensions

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

Views

Author

N. J. A. Sloane

Keywords

References

  • 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

Crossrefs

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

Programs

  • PARI
    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

Formula

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

Extensions

More terms from Christian G. Bower, Nov 15 1998

A035349 "DIK" (bracelet, indistinct, unlabeled) transform of A000237.

Original entry on oeis.org

1, 1, 2, 5, 14, 43, 143, 496, 1794, 6667, 25345, 98032, 384713, 1527480, 6125327, 24770186, 100897860, 413595904, 1704840125, 7062024986, 29382224119, 122731488819, 514491387498, 2163757816681, 9126920239124, 38602653740841
Offset: 0

Views

Author

Christian G. Bower, Nov 15 1998

Keywords

Links

Crossrefs

Cf. A000083, A000237, A000314, A035082, A035350-A035357.

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

Views

Author

Christian G. Bower, Nov 15 1998

Keywords

Links

Crossrefs

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

Formula

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

A035085 Number of polygonal cacti (Husimi graphs) with n nodes.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 2, 5, 7, 16, 28, 63, 131, 301, 673, 1600, 3773, 9158, 22319, 55255, 137563, 345930, 874736, 2227371, 5700069, 14664077, 37888336, 98310195, 256037795, 669184336, 1754609183, 4614527680
Offset: 0

Views

Author

Christian G. Bower, Nov 15 1998

Keywords

References

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

Links

Crossrefs

Cf. A035082, A035083, A035084.

Programs

  • PARI
    BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}
DIK(p,n)={(sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d))) + ((1+p)^2/(1-subst(p, x, x^2))-1)/2)/2}
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(p=O(x)); for(n=1, n, p=x+x^2*Ser(EulerT(Vec(BIK(p)-1)-Vec(p)))); Vec(1 + DIK(p, n) - (p^2 + subst(p, x, x^2))/2 - p*(BIK(p)-1-p))} \\ Andrew Howroyd, Aug 31 2018

Formula

G.f.: A(x) = B(x) + C(x) - B(x)*D(x) where B, C, D are g.f.s of A035082, A035083, A035084, respectively.

Extensions

Terms a(32) and beyond from Andrew Howroyd, Aug 31 2018

A035088 Number of labeled polygonal cacti (Husimi graphs) with n nodes.

Original entry on oeis.org

1, 1, 0, 1, 3, 27, 240, 2985, 42840, 731745, 14243040, 313570845, 7683984000, 207685374435, 6135743053440, 196754537704725, 6805907485977600, 252620143716765825, 10015402456976716800, 422410127508300756825, 18884777200534941696000
Offset: 0

Views

Author

Christian G. Bower, Nov 15 1998

Keywords

Comments

A Husimi tree is a connected graph in which no line lies on more than one cycle [Harary, 1953]. - Jonathan Vos Post, Mar 12 2010

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301.
  • F. Harary and R. Z. Norman "The Dissimilarity Characteristic of Husimi Trees" Annals of Mathematics, 58 1953, pp. 134-141.
  • F. Harary and E. M. Palmer, Graphical Enumeration, p. 71.
  • F. Harary and G. E. Uhlenbeck "On the Number of Husimi Trees" Proc. Nat. Acad. Sci. USA vol. 39. pp. 315-322, 1953.
  • F. Harary, G. Uhlenbeck (1953), "On the number of Husimi trees, I", Proceedings of the National Academy of Sciences 39: 315-322. - From Jonathan Vos Post, Mar 12 2010

Links

Crossrefs

Cf. A035082, A035083, A035084, A035085, A035086, A035087.

Programs

  • Mathematica
    max = 20; s = 1+InverseSeries[Series[E^(x^2/(2*(x-1)))*x, {x, 0, max}], x]; a[n_] := SeriesCoefficient[s, n]*(n-1)!; a[0]=1; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Feb 27 2016, after Vaclav Kotesovec at A035087 *)

Formula

a(n) = A035087(n)/n, n > 0.

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

Views

Author

Christian G. Bower, Nov 15 1998

Keywords

Links

Crossrefs

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

Programs

  • PARI
    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

Formula

Shifts left under transform T where Ta = WEIGH(BHK(a)).
Showing 1-10 of 18 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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