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 17 results. Next

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)).

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)).

A380632 Number of simple connected graphs on n unlabeled nodes with each node a member of exactly one cycle.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 2, 3, 5, 9, 14, 28, 49, 95, 182, 369, 733, 1509, 3103, 6504, 13627, 28949, 61701, 132457, 285454, 618863, 1346022, 2940287, 6444364, 14172744, 31257883, 69142445, 153333476, 340880766, 759549740, 1696122213, 3795178540, 8508326129, 19109193805, 42991993545, 96881110654
Offset: 0

Views

Author

Andrew Howroyd, Feb 24 2025

Keywords

Comments

All such graphs are cactus graphs (with bridges allowed).

Links

Crossrefs

Row sums of A380631.
Cf. A000083, A380805.

Programs

  • PARI
    Vec(G(40)) \\ G() defined in A380631.

A380805 Number of unlabeled simple connected graphs with n nodes of degree at most 3 and each node a member of exactly one cycle.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 2, 3, 4, 7, 10, 18, 27, 49, 81, 147, 256, 476, 858, 1612, 2991, 5676, 10729, 20575, 39423, 76232, 147602, 287518, 561195, 1100190, 2161552, 4261059, 8418035, 16675006, 33098322, 65844566, 131233923, 262066375, 524224509, 1050414569
Offset: 0

Views

Author

Gordon Hamilton, Feb 23 2025

Keywords

Comments

All such graphs are cactus graphs (with bridges allowed).

Links

Crossrefs

Row sums of A380633.
Cf. A000083, A001349, A317722, A380632 (nodes of any degree).

Programs

Extensions

a(13) onwards from Andrew Howroyd, Feb 24 2025

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

Views

Author

Christian G. Bower, Nov 15 1998

Keywords

Links

Crossrefs

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

A332650 Number of polygonal cacti on 2n-1 unlabeled nodes with every polygon having an odd prime number of edges.

Original entry on oeis.org

1, 1, 2, 4, 10, 30, 105, 400, 1654, 7229, 32944, 154749, 744973, 3655993, 18232812, 92162974, 471301437, 2434542190, 12687850499, 66646225443, 352548333438, 1876770716627, 10048289587337, 54079948967654, 292447643655469, 1588388448970674, 8661869330014601
Offset: 1

Views

Author

Andrew Howroyd, Feb 18 2020

Keywords

Examples

			a(3) = 2 because there are two cacti on 5 nodes which are a pentagon and 2 triangles joined at a node.

Links

Crossrefs

Cf. A000083, A035085, A091487, A332649, A332651.

Programs

  • PARI
    \\ Here UCacti gives number of unrooted cacti with restricted polygons.
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
RCacti(u)={my(v=[1]); while(#v<#u, my(g=x*Ser(v), g2=subst(g,x,x^2) + O(x^2*x^#v), r=sum(k=1, #u-1, my(c=u[k+1]); if(c, c*(g^k + g^(k%2)*g2^(k\2))))/2 + O(x^#u)); v=concat([1], EulerT(Vec(r, 1-serprec(r, x))))); v}
UCacti(u)={my(p=x*Ser(RCacti(u))); my(g(d)=subst(p + O(x*x^(#u\d)), x, x^d)); Vec(g(1) + sum(k=1, #u, my(c=u[k]); if(c, sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/(2*k) - (g(1)^k)/2 + if(k%2==0, g(2)^(k/2) - g(1)^2*g(2)^(k/2-1))/4)))}
seq(n)={my(v=UCacti(vector(2*n-1, i, i>2 && isprime(i)))); vector(n, i, v[2*i-1])}

A332651 Number of polygonal cacti on n unlabeled nodes with every polygon having an even number of edges.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 4, 2, 7, 9, 14, 26, 48, 71, 154, 243, 478, 894, 1631, 3149, 6062, 11295, 22469, 42900, 83528, 164829, 321012, 632960, 1255613, 2472803, 4928140, 9808439, 19533534, 39134059, 78345317, 157177556, 316398963, 636790282, 1284910954
Offset: 0

Views

Author

Andrew Howroyd, Feb 18 2020

Keywords

Comments

Bridges are disallowed.

Examples

			a(6) = 1 corresponding with a hexagon.
a(7) = 1 corresponding with two quadrilaterals joined at a node.

Links

Crossrefs

Cf. A000083, A035085, A091487, A332649, A332650.

Programs

  • PARI
    \\ See A332650 for UCacti.
seq(n)={concat([1], UCacti(vector(n, i, i>2&&i%2==0)))}
Showing 1-10 of 17 results. Next

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

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