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.

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A000299 Number of n-node rooted trees of height 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 13, 36, 93, 225, 528, 1198, 2666, 5815, 12517, 26587, 55933, 116564, 241151, 495417, 1011950, 2055892, 4157514, 8371318, 16792066, 33564256, 66875221, 132849983, 263192599, 520087551, 1025295487, 2016745784, 3958608430, 7754810743
Offset: 1

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

Column h=4 of A034781.

Programs

  • Maple
    For Maple program see link in A000235.
  • Mathematica
    f[n_] := Nest[CoefficientList[Series[Product[1/(1 - x^i)^#[[i]], {i, 1, Length[#]}], {x, 0, 40}], x] &, {1}, n];f[4]-f[3] (* Geoffrey Critzer, Aug 01 2013 *)

Formula

Cf. A001384-A001383. - Christian G. Bower

A000342 Number of n-node rooted trees of height 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 5, 19, 61, 180, 498, 1323, 3405, 8557, 21103, 51248, 122898, 291579, 685562, 1599209, 3705122, 8532309, 19543867, 44552066, 101124867, 228640542, 515125815, 1156829459, 2590247002, 5784031485, 12883390590, 28629914457
Offset: 1

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

Column h=5 of A034781.

Programs

  • Maple
    For Maple program see link in A000235.
  • Mathematica
    f[n_] := Nest[CoefficientList[Series[Product[1/(1 - x^i)^#[[i]], {i, 1, Length[#]}], {x, 0, 40}], x] &, {1}, n];f[5]-f[4] (* Geoffrey Critzer, Aug 01 2013 *)
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1 || k<1, 0, Sum[ Binomial[ b[i-1, i-1, k-1]+j-1, j]*b[n-i*j, i-1, k], {j, 0, n/i}]]]; a[n_] := b[n- 1, n-1, 5] - b[n-1, n-1, 4]; Array[a, 40] (* Jean-François Alcover, Feb 07 2016, after Alois P. Heinz in A034781 *)

Formula

A001385-A001384. (Christian G. Bower).

A000473 Number of genus 0 rooted maps with 5 faces and n vertices.

Original entry on oeis.org

14, 386, 5868, 65954, 614404, 5030004, 37460376, 259477218, 1697186964, 10596579708, 63663115880, 370293754740, 2095108370600, 11574690111400, 62629794691632, 332742342741090, 1739371969822260, 8961709528660140, 45576855706440520, 229087231033907708
Offset: 4

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).
  • T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

Links

Crossrefs

Column 5 of A269920.
Column 0 of A270409.

Programs

  • Mathematica
    CoefficientList[(1/x)(1-Sqrt[1-4x])(17+16x-(10+4x)Sqrt[1-4x])/(1-4x)^(11/2) + O[x]^36, x] (* Jean-François Alcover, Feb 08 2016 *)
  • PARI
    seq(n)={my(g=sqrt(1-4*x + O(x*x^n))); Vec((1-g)*(17+16*x-(10+4*x)*g)/((1-4*x)^5*g))} \\ Andrew Howroyd, Mar 28 2021

Formula

G.f.: x^3*(1-sqrt(1-4*x))*(17+16*x-(10+4*x)*sqrt(1-4*x))/(1-4*x)^(11/2). - Sean A. Irvine, Nov 14 2010

Extensions

More terms from Sean A. Irvine, Nov 14 2010

A000502 Number of genus 0 rooted maps with 6 faces and n vertices.

Original entry on oeis.org

42, 1586, 31388, 442610, 5030004, 49145460, 429166584, 3435601554, 25658464260, 181055975100, 1218601601672, 7880146275092, 49238911113224, 298652277299880, 1764885293279472, 10192638073849554, 57674223198273444, 320430129184331628, 1751190732477786600, 9428906326013866076
Offset: 5

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).
  • T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

Links

Crossrefs

Column 6 of A269920.
Column 0 of A270410.

Programs

  • Mathematica
    CoefficientList[ x(1-Sqrt[1-4x])(105+92x-(84+76x)Sqrt[1-4x])/(1-4x)^7/x^2 + O[x]^30, x] (* Jean-François Alcover, Feb 09 2016 *)
  • PARI
    seq(n)={my(g=sqrt(1-4*x + O(x*x^n))); Vec((1-g)*(105+92*x - (84+76*x)*g)/((1-4*x)^7))} \\ Andrew Howroyd, Mar 28 2021

Formula

G.f.: x^4*(1-sqrt(1-4*x))*(105+92*x-(84+76*x)*sqrt(1-4*x))/(1-4*x)^7. - Sean A. Irvine, Nov 14 2010

Extensions

More terms from Sean A. Irvine, Nov 14 2010

A006445 Number of n-edge 3-connected planar maps with a sense-reversing automorphism.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 2, 1, 4, 9, 12, 27, 50, 99, 188, 386, 740, 1528, 3012, 6192, 12376, 25594, 51628, 107135, 218100, 453895, 930812, 1943281, 4009512, 8394915, 17414512
Offset: 1

Views

Author

N. J. A. Sloane, T. R. S. Walsh, personal communication

Keywords

Comments

Maps are counted up to isomorphism which is not restricted to those that preserve the orientation of the sphere.

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • Computed by Nick Wormald (unpublished).

Links

A297193 Irregular triangle read by rows: rows are partial alternating sums of rows of A297191.

Original entry on oeis.org

1, 1, 4, 1, 1, 8, 17, 8, 1, 1, 12, 49, 80, 49, 12, 1, 1, 16, 97, 280, 401, 280, 97, 16, 1, 1, 20, 161, 672, 1569, 2084, 1569, 672, 161, 20, 1, 1, 24, 241, 1320, 4321, 8752, 11073, 8752, 4321, 1320, 241, 24, 1, 1, 28, 337, 2288, 9681, 26684, 48833, 59712, 48833, 26684, 9681
Offset: 0

Views

Author

N. J. A. Sloane, Jan 10 2018

Keywords

Examples

			Triangle begins:
1,
1,4,1,
1,8,17,8,1,
1,12,49,80,49,12,1,
1,16,97,280,401,280,97,16,1,
1,20,161,672,1569,2084,1569,672,161,20,1,
...

Links

Crossrefs

Cf. A008288, A297191, A297194.
The middle and next to middle columns are A089165 and A089383.

Programs

A355671 Number of labeled trees on [n] that are bicentered.

Original entry on oeis.org

0, 0, 1, 0, 12, 60, 570, 8190, 134456, 2408616, 49307670, 1159112130, 30619757652, 891045909468, 28244653953698, 969331283419590, 35858099428919280, 1423688804991442896, 60402176709135347502, 2726896792761748601226, 130498364319404393167820
Offset: 0

Views

Author

Geoffrey Critzer, Aug 02 2022

Keywords

Comments

This is the labeled version of A000677 where pertinent definitions can be found.

Links

Crossrefs

Cf. A034854, A000677, A356292, A000272.

Programs

  • Mathematica
    nn = 20; T = NestList[z Exp[#] &, z, nn]; G[k_, z_] := T[[k + 1]];
H[k_, z_] := T[[k + 1]] - T[[k]];H[0, z_] := z; ReplacePart[ Sum[Range[0, nn]! CoefficientList[Series[H[m, z]^2/2, {z, 0, nn}], z], {m, 1, nn/2 - 1}], 3 -> 1]

Formula

a(n) = Sum_{odd d} A034854(n,d).
a(n) = A000272(n) - A356292(n).

A356292 Number of labeled trees on [n] that are centered.

Original entry on oeis.org

1, 1, 0, 3, 4, 65, 726, 8617, 127688, 2374353, 50692330, 1198835561, 31297606572, 901114484569, 28449258421598, 976863784939785, 36199494609008656, 1438734246518372897, 61037354387458904274, 2753490065023053584713, 131645635680595606832180
Offset: 0

Views

Author

Geoffrey Critzer, Aug 02 2022

Keywords

Comments

This is the labeled version of A000676 which has the pertinent definitions.

Links

Crossrefs

Cf. A034854, A000676, A000272, A355671.

Programs

  • Mathematica
    nn = 20; T = NestList[z Exp[#] &, z, nn]; G[k_, z_] := T[[k + 1]];H[k_, z_] := T[[k + 1]] - T[[k]];H[0, z_] := z; ReplacePart[ Sum[Range[0, nn]!CoefficientList[Series[G[m, z] (Exp[H[m, z]] - 1 - H[m, z]), {z, 0, nn}], z], {m, 0, nn/2 - 2}], {1 -> 1, 2 -> 1}]

Formula

a(n) = Sum_{d even} A034854(n,d).
a(n) = A000272(n) - A355671(n)

A379432 Triangle read by rows: T(n,k) is the number of unsensed 2-connected (nonseparable) planar maps with n edges and k vertices, n >= 2, 2 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 7, 3, 1, 1, 4, 13, 13, 4, 1, 1, 5, 29, 44, 29, 5, 1, 1, 7, 51, 139, 139, 51, 7, 1, 1, 8, 92, 370, 623, 370, 92, 8, 1, 1, 10, 147, 913, 2307, 2307, 913, 147, 10, 1, 1, 12, 240, 2048, 7644, 11673, 7644, 2048, 240, 12, 1, 1, 14, 357, 4295, 22344, 50174, 50174, 22344, 4295, 357, 14, 1
Offset: 2

Views

Author

Andrew Howroyd, Jan 14 2025

Keywords

Comments

The maps considered here may include parallel edges.
The number of faces is n + 2 - k.

Examples

			Triangle begins:
   1;
   1,  1;
   1,  1,   1;
   1,  2,   2,   1;
   1,  3,   7,   3,    1;
   1,  4,  13,  13,    4,    1;
   1,  5,  29,  44,   29,    5,   1;
   1,  7,  51, 139,  139,   51,   7,   1;
   1,  8,  92, 370,  623,  370,  92,   8,  1;
   1, 10, 147, 913, 2307, 2307, 913, 147, 10, 1;
   ...

Links

Crossrefs

Row sums are A006403.
Cf. A082680 (rooted), A342061 (sensed), A212438 (3-connected), A277741, A342060.

Formula

T(n,k) = T(n, n+2-k).

A001852 Total diameter of labeled trees with n nodes.

Original entry on oeis.org

0, 1, 6, 44, 430, 5322, 79184, 1381144, 27730602, 630422390, 16006336852, 448982630340, 13792542282974, 460632431511826, 16620059192605080, 644338908974954672, 26713929408696716242, 1179487563859389821166
Offset: 1

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. A034854.

Formula

a(1) = 0, a(2) = 1, a(n) = Sum_{k=2..n-1} A034854(n,k)*k. - Sean A. Irvine, Mar 24 2016

Extensions

More terms from Sean A. Irvine, Mar 24 2016
Previous Showing 31-40 of 47 results. Next

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

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