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 1-5 of 5 results.

A269924 Triangle read by rows: T(n,f) is the number of rooted maps with n edges and f faces on an orientable surface of genus 4.

Original entry on oeis.org

225225, 12317877, 12317877, 351683046, 792534015, 351683046, 7034538511, 26225260226, 26225260226, 7034538511, 111159740692, 600398249550, 993494827480, 600398249550, 111159740692, 1480593013900, 10743797911132, 25766235457300, 25766235457300, 10743797911132, 1480593013900, 17302190625720, 160576594766588, 517592962672296, 750260619502310, 517592962672296, 160576594766588, 17302190625720
Offset: 8

Views

Author

Gheorghe Coserea, Mar 15 2016

Keywords

Comments

Row n contains n-7 terms.

Examples

			Triangle starts:
n\f  [1]           [2]           [3]           [4]
[8]  225225;
[9]  12317877,     12317877;
[10] 351683046,    792534015,    351683046;
[11] 7034538511,   26225260226,  26225260226,  7034538511;
[12] ...

Links

Crossrefs

Columns f=1-10 give: A288271 f=1, A288272 f=2, A288273 f=3, A288274 f=4, A288275 f=5, A288276 f=6, A288277 f=7, A288278 f=8, A288279 f=9, A288280 f=10.
Cf. A035309, A269921, A269922, A269923, A269925, A270406, A270407, A270408, A270409, A270410, A270412.
Row sums give A215402 (column 4 of A269919).

Programs

  • Mathematica
    Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n<0 || f<0 || g<0 = 0;
Q[n_, f_, g_] := Q[n, f, g] = 6/(n+1)((2n-1)/3 Q[n-1, f, g] + (2n-1)/3 Q[n - 1, f-1, g] + (2n-3)(2n-2)(2n-1)/12 Q[n-2, f, g-1] + 1/2 Sum[l = n-k; Sum[v = f-u; Sum[j = g-i; Boole[l >= 1 && v >= 1 && j >= 0] (2k-1) (2l-1) Q[k-1, u, i] Q[l-1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
Table[Q[n, f, 4], {n, 8, 14}, {f, 1, n-7}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)
  • PARI
    N = 14; G = 4; gmax(n) = min(n\2, G);
Q = matrix(N + 1, N + 1);
Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };
Qset(n, g, v) = { Q[n+1, g+1] = v };
Quadric({x=1}) = {
  Qset(0, 0, x);
  for (n = 1, length(Q)-1, for (g = 0, gmax(n),
    my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),
       t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),
       t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,
       (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));
    Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));
};
Quadric('x);
concat(apply(p->Vecrev(p/'x), vector(N+1 - 2*G, n, Qget(n-1 + 2*G, G))))

A269925 Triangle read by rows: T(n,f) is the number of rooted maps with n edges and f faces on an orientable surface of genus 5.

Original entry on oeis.org

59520825, 4304016990, 4304016990, 158959754226, 354949166565, 158959754226, 4034735959800, 14805457339920, 14805457339920, 4034735959800, 79553497760100, 420797306522502, 691650582088536, 420797306522502, 79553497760100, 1302772718028600, 9220982517965400, 21853758736216200, 21853758736216200, 9220982517965400, 1302772718028600
Offset: 10

Views

Author

Gheorghe Coserea, Mar 15 2016

Keywords

Comments

Row n contains n-9 terms.

Examples

			Triangle starts:
n\f  [1]             [2]             [3]             [4]
[10] 59520825;
[11] 4304016990,     4304016990;
[12] 15895975422,    354949166565,   158959754226;
[13] 4034735959800,  14805457339920, 14805457339920, 4034735959800;
[14] ...

Links

Crossrefs

Rooted maps of genus 5 with n edges and f faces for 1<=f<=10: A288281 f=1, A288282 f=2, A288283 f=3, A288284 f=4, A288285 f=5, A288286 f=6, A288287 f=7, A288288 f=8, A288289 f=9, A288290 f=10.
Row sums give A238355 (column 5 of A269919).
Cf. A035309, A269921, A269922, A269923, A269924, A270406, A270407, A270408, A270409, A270410, A270411, A270412.

Programs

  • Mathematica
    Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n<0 || f<0 || g<0 = 0;
Q[n_, f_, g_] := Q[n, f, g] = 6/(n+1)((2n-1)/3 Q[n-1, f, g] + (2n-1)/3 Q[n - 1, f-1, g] + (2n-3)(2n-2)(2n-1)/12 Q[n-2, f, g-1] + 1/2 Sum[l = n-k; Sum[v = f-u; Sum[j = g-i; Boole[l >= 1 && v >= 1 && j >= 0] (2k-1)(2l-1) Q[k-1, u, i] Q[l-1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
Table[Q[n, f, 5], {n, 10, 15}, {f, 1, n-9}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)
  • PARI
    N = 15; G = 5; gmax(n) = min(n\2, G);
Q = matrix(N + 1, N + 1);
Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };
Qset(n, g, v) = { Q[n+1, g+1] = v };
Quadric({x=1}) = {
  Qset(0, 0, x);
  for (n = 1, length(Q)-1, for (g = 0, gmax(n),
    my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),
       t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),
       t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,
       (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));
    Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));
};
Quadric('x);
concat(apply(p->Vecrev(p/'x), vector(N+1 - 2*G, n, Qget(n-1 + 2*G, G))))

A288084 a(n) is the number of rooted maps with n edges and 4 faces on an orientable surface of genus 2.

Original entry on oeis.org

66066, 3288327, 85421118, 1558792200, 22555934280, 276221817810, 2979641557620, 29079129795702, 261637840342860, 2200626948631386, 17486142956133684, 132344695964811720, 960323177351524512, 6716133365837116980, 45466867668336614472, 299027167905149145858, 1916387674555902480660
Offset: 7

Views

Author

Gheorghe Coserea, Jun 05 2017

Keywords

Links

Crossrefs

Rooted maps of genus 2 with n edges and f faces for 1<=f<=10: A006298 f=1, A288082 f=2, A288083 f=3, this sequence, A288085 f=5, A288086 f=6, A288087 f=7, A288088 f=8, A288089 f=9, A288090 f=10.
Column 4 of A269922, column 2 of A270408.
Cf. A000108.

Programs

  • Mathematica
    Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
Q[n_, f_, g_] := Q[n, f, g] = 6/(n + 1) ((2 n - 1)/3 Q[n - 1, f, g] + (2 n - 1)/3 Q[n - 1, f - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 Q[n - 2, f, g - 1] + 1/2 Sum[l = n - k; Sum[v = f - u; Sum[j = g - i; Boole[l >= 1 && v >= 1 && j >= 0] (2 k - 1) (2 l - 1) Q[k - 1, u, i] Q[l - 1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
a[n_] := Q[n, 4, 2];
Table[a[n], {n, 7, 23}] (* Jean-François Alcover, Oct 18 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288084_ser(N) = {
  my(y = A000108_ser(N+1));
  3*y*(y-1)^7*(9318*y^6 + 178328*y^5 + 177929*y^4 - 611583*y^3 + 195218*y^2 + 110388*y - 37576)/(y-2)^20;
};
Vec(A288084_ser(17))

A270409 Triangle read by rows: T(n,g) is the number of rooted maps with n edges and 5 faces on an orientable surface of genus g.

Original entry on oeis.org

14, 386, 5868, 2310, 65954, 100156, 614404, 2278660, 570570, 5030004, 36703824, 34374186, 37460376, 472592916, 1059255456, 211083730, 259477218, 5188948072, 22555934280, 16476937840, 1697186964, 50534154408, 375708427812, 647739636160, 111159740692
Offset: 4

Views

Author

Gheorghe Coserea, Mar 17 2016

Keywords

Comments

Row n contains floor((n-2)/2) terms.

Examples

			Triangle starts:
n\g    [0]           [1]           [2]           [3]
[4]    14;
[5]    386;
[6]    5868,         2310;
[7]    65954,        100156;
[8]    614404,       2278660,      570570;
[9]    5030004,      36703824,     34374186;
[10]   37460376,     472592916,    1059255456,   211083730;
[11]   259477218,    5188948072,   22555934280,  16476937840;
[12]   ...

Links

Crossrefs

Cf. A270408.

Programs

  • Mathematica
    Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
Q[n_, f_, g_] := Q[n, f, g] = 6/(n+1) ((2n-1)/3 Q[n-1, f, g] + (2n-1)/3 Q[n - 1, f-1, g] + (2n-3) (2n-2) (2n-1)/12 Q[n-2, f, g-1] + 1/2 Sum[l = n-k; Sum[v = f-u; Sum[j = g-i; Boole[l >= 1 && v >= 1 && j >= 0] (2k-1) (2l-1) Q[k - 1, u, i] Q[l - 1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
T[n_, g_] := Q[n, 5, g];
Table[T[n, g], {n, 4, 12}, {g, 0, Quotient[n-2, 2]-1}] // Flatten (* Jean-François Alcover, Oct 18 2018 *)
  • PARI
    N = 12; F = 5; gmax(n) = n\2;
Q = matrix(N + 1, N + 1);
Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };
Qset(n, g, v) = { Q[n+1, g+1] = v };
Quadric({x=1}) = {
  Qset(0, 0, x);
  for (n = 1, length(Q)-1, for (g = 0, gmax(n),
    my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),
       t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),
       t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,
       (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));
    Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));
};
Quadric('x + O('x^(F+1)));
v = vector(N+2-F, n, vector(1 + gmax(n-1), g, polcoeff(Qget(n+F-2, g-1), F)));
concat(v)

A000365 Number of genus 0 rooted planar maps with 4 faces and n vertices.

Original entry on oeis.org

5, 93, 1030, 8885, 65954, 442610, 2762412, 16322085, 92400330, 505403910, 2687477780, 13957496098, 71053094420, 355548314180, 1752827693528, 8529176056965, 41026491589722, 195327793313790, 921451498774660, 4311086414580022, 20019238138410940
Offset: 3

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 4 of A269920.
Column 0 of A270408.

Programs

  • Mathematica
    nn = 20; CoefficientList[Series[x^2 (1 - Sqrt[1 - 4 x]) (7 + 4 x - 2 Sqrt[1 - 4 x])/(2 (4 x - 1)^4), {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *)
  • PARI
    seq(n)={my(g=sqrt(1-4*x + O(x*x^n))); Vec((1-g)*(7+4*x-2*g)/(2*(1-4*x)^4))} \\ Andrew Howroyd, Mar 27 2021

Formula

G.f.: x^2*(1-sqrt(1-4*x))*(7+4*x-2*sqrt(1-4*x))/(2*(4*x-1)^4). - corrected for right offset by Vaclav Kotesovec, Aug 13 2013
a(n) ~ n^3*4^n/24 * (1-4/(sqrt(Pi*n))). - Vaclav Kotesovec, Aug 13 2013

Extensions

More terms from Sean A. Irvine, Nov 14 2010
Showing 1-5 of 5 results.

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

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