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 1-7 of 7 results.

A342985 Triangle read by rows: T(n,k) is the number of tree-rooted loopless planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 0, 3, 3, 0, 0, 4, 36, 4, 0, 0, 5, 135, 135, 5, 0, 0, 6, 360, 1368, 360, 6, 0, 0, 7, 798, 7350, 7350, 798, 7, 0, 0, 8, 1568, 28400, 73700, 28400, 1568, 8, 0, 0, 9, 2826, 89073, 474588, 474588, 89073, 2826, 9, 0, 0, 10, 4770, 241220, 2292790, 4818092, 2292790, 241220, 4770, 10, 0
Offset: 0

Views

Author

Andrew Howroyd, Apr 03 2021

Keywords

Comments

The number of vertices is n + 2 - k.
For k >= 2, column k without the initial zero term is a polynomial of degree 4*(k-2)+1.

Examples

			Triangle begins:
  1;
  0, 0;
  0, 2,    0;
  0, 3,    3,     0;
  0, 4,   36,     4,     0;
  0, 5,  135,   135,     5,     0;
  0, 6,  360,  1368,   360,     6,    0;
  0, 7,  798,  7350,  7350,   798,    7, 0;
  0, 8, 1568, 28400, 73700, 28400, 1568, 8, 0;
  ...

Links

Crossrefs

Columns and diagonals 3..5 are A006428, A006429, A006430.
Row sums are A342986.
Cf. A342980, A342982, A342984, A342987.

Programs

  • PARI
    \\ here G(n,y) is A342984 as g.f.
F(n,y)={sum(n=0, n, x^n*sum(i=0, n, my(j=n-i); y^i*(2*i+2*j)!/(i!*(i+1)!*j!*(j+1)!))) + O(x*x^n)}
G(n,y)={my(g=F(n,y)); subst(g, x, serreverse(x*g^2))}
H(n)={my(g=G(n, y)-x*(1+y), v=Vec(sqrt(serreverse(x/g^2)/x))); vector(#v, n, Vecrev(v[n], n))}
{ my(T=H(8)); for(n=1, #T, print(T[n])) }

Formula

T(n,n+2-k) = T(n,k).
G.f.: A(x,y) satisfies A(x,y) = G(x*A(x,y)^2,y) where G(x,y) + x*(1+y) is the g.f. of A342984.

A342984 Triangle read by rows: T(n,k) is the number of nonseparable tree-rooted planar maps with n edges and k faces, n >= 0, k = 1..n+1.

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 0, 3, 3, 0, 0, 4, 20, 4, 0, 0, 5, 75, 75, 5, 0, 0, 6, 210, 604, 210, 6, 0, 0, 7, 490, 3150, 3150, 490, 7, 0, 0, 8, 1008, 12480, 27556, 12480, 1008, 8, 0, 0, 9, 1890, 40788, 170793, 170793, 40788, 1890, 9, 0, 0, 10, 3300, 115500, 829920, 1565844, 829920, 115500, 3300, 10, 0
Offset: 0

Views

Author

Andrew Howroyd, Apr 03 2021

Keywords

Comments

The number of vertices is n + 2 - k.
A tree-rooted planar map is a planar map with a distinguished spanning tree.
For k >= 2, column k is a polynomial of degree 4*(k-2)+1.

Examples

			Triangle begins:
  1;
  1, 1;
  0, 2,    0;
  0, 3,    3,     0;
  0, 4,   20,     4,     0;
  0, 5,   75,    75,     5,     0;
  0, 6,  210,   604,   210,     6,    0;
  0, 7,  490,  3150,  3150,   490,    7, 0;
  0, 8, 1008, 12480, 27556, 12480, 1008, 8, 0;
  ...

Links

Crossrefs

Columns (and diagonals) 3..5 are A006411, A006412, A006413.
Row sums are A004304.
Cf. A342982, A342985, A342987.

Programs

  • PARI
    \\ here F(n,y) gives A342982 as g.f.
F(n,y)={sum(n=0, n, x^n*sum(i=0, n, my(j=n-i); y^i*(2*i+2*j)!/(i!*(i+1)!*j!*(j+1)!))) + O(x*x^n)}
H(n)={my(g=F(n,y), v=Vec(subst(g, x, serreverse(x*g^2)))); vector(#v, n, Vecrev(v[n], n))}
{ my(T=H(8)); for(n=1, #T, print(T[n])) }

Formula

T(n,n+2-k) = T(n,k).
G.f. A(x,y) satisfies F(x,y) = A(x*F(x,y)^2,y) where F(x,y) is the g.f. of A342982.

A006470 Number of tree-rooted planar maps with 3 faces and n vertices and no isthmuses.

Original entry on oeis.org

2, 15, 60, 175, 420, 882, 1680, 2970, 4950, 7865, 12012, 17745, 25480, 35700, 48960, 65892, 87210, 113715, 146300, 185955, 233772, 290950, 358800, 438750, 532350, 641277, 767340, 912485, 1078800, 1268520, 1484032, 1727880, 2002770, 2311575, 2657340, 3043287, 3472820, 3949530, 4477200, 5059810, 5701542, 6406785, 7180140
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) is the number of ordered rooted trees with n+3 non-root nodes that have 3 leaves; see A108838. - Joerg Arndt, Aug 18 2014

References

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

Links

Crossrefs

Column 3 of A342987.
Cf. A027789, A108838.

Programs

Formula

a(n) = (n+1)*binomial(n+3, 4).
a(n) = A027789(n)/2.
From Zerinvary Lajos, Dec 14 2005: (Start)
a(n) = binomial(n+2, 2)*binomial(n+4, 3)/2;
G.f.: x*(2+3*x)/(1-x)^6. (End)
From Wesley Ivan Hurt, May 02 2015: (Start)
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
a(n) = n*(n+1)^2*(n+2)*(n+3)/24. (End)
Sum_{n>=1} 1/a(n) = 61/3 - 2*Pi^2. - Jaume Oliver Lafont, Jul 15 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2 - 16*log(2) + 5/3. - Amiram Eldar, Jan 28 2022

Extensions

Name clarified by Andrew Howroyd, Apr 03 2021

A342988 Number of tree-rooted planar maps with n edges and no isthmuses.

Original entry on oeis.org

1, 1, 4, 23, 162, 1292, 11214, 103497, 1000810, 10039100, 103725188, 1098151276, 11866435816, 130477138014, 1456320910090, 16468167354971, 188369396046810, 2176619115192140, 25379588118629856, 298341351434460488, 3532848638781046852, 42113699799069958732
Offset: 0

Views

Author

Andrew Howroyd, Apr 03 2021

Keywords

Links

Crossrefs

Row sums of A342987.
Cf. A004304, A342986.

Programs

  • PARI
    \\ here J(n) gives A004304 as g.f.
J(n)={my(g=(1-sqrt(1-4*x+O(x^3*x^n)))/(2*x), h=serconvol(g, (g-1)/x)); sqrt(x/serreverse(x*h^2))}
seq(n)={my(g=J(n)-x, p=O(1)); while(serprec(p, x)<=n, p = subst(g, x, x*p^2)); Vec(p)}

Formula

G.f.: A(x) satisfies A(x) = G(x*A(x)^2) where G(x) + x is the g.f. of A004304.

A006432 Number of tree-rooted planar maps with 3 vertices and n faces and no isthmuses.

Original entry on oeis.org

0, 3, 60, 650, 5352, 37681, 239752, 1421226, 7996160, 43219990, 226309800, 1154900708, 5769562736, 28312118565, 136830224464, 652656300122, 3077631550512, 14367512295274, 66478236840680, 305161336656876, 1390869368495728, 6298727501142218, 28358908010334960
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

A diagonal of A342987.

Formula

a(n) seems to be divisible by n+1. - Ralf Stephan, Sep 01 2003

Extensions

Name clarified and terms a(13) and beyond from Andrew Howroyd, Apr 06 2021

A006433 Number of tree-rooted planar maps with 4 vertices and n faces and no isthmuses.

Original entry on oeis.org

0, 4, 175, 3324, 42469, 429120, 3711027, 28723640, 204598130, 1366223880, 8664086470, 52673351080, 309164754285, 1761471681568, 9783594370723, 53154274959360, 283267669144390, 1484104565936920, 7658877239935362, 38993558097982312, 196127054929939810
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

A map without isthmuses can also be called bridgeless.

References

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

Links

Crossrefs

A diagonal of A342987.

Extensions

Terms a(12) and beyond from Andrew Howroyd, Apr 06 2021

A006471 Number of tree-rooted planar maps with 4 faces and n vertices and no isthmuses.

Original entry on oeis.org

5, 84, 650, 3324, 13020, 42240, 118998, 300300, 693693, 1490060, 3011580, 5779592, 10608000, 18728832, 31957620, 52907400, 85261341, 134115300, 206402966, 311417700, 461446700, 672534720, 965396250, 1366496820, 1909325925, 2635885980, 3598423704, 4861432400, 6503955744, 8622225920
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

Column 4 of A342987.

Programs

  • Mathematica
    A006471[n_] := n*(n + 1)*(n + 2)^2*(n + 3)*(n + 4)*(n + 5)*(n*(13*n + 73) + 54)/60480;
Array[A006471, 50] (* Paolo Xausa, Aug 20 2025 *)
  • PARI
    a(n) = (n*(2+n)^2*(3240 +10158*n +11777*n^2 +6400*n^3 +1770*n^4 +242*n^5 +13*n^6))/60480 \\ Andrew Howroyd, Apr 03 2021

Formula

From Colin Barker, Apr 09 2013: (Start)
a(n) = (n*(2+n)^2*(3240 + 10158*n + 11777*n^2 + 6400*n^3 + 1770*n^4 + 242*n^5 + 13*n^6))/60480.
G.f.: x*(4*x^3 + 35*x^2 + 34*x + 5) / (x-1)^10. (End)

Extensions

Name clarified and terms a(12) and beyond from Andrew Howroyd, Apr 03 2021
Showing 1-7 of 7 results.

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

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