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 10 results.

A269923 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 3.

Original entry on oeis.org

1485, 56628, 56628, 1169740, 2668750, 1169740, 17454580, 66449432, 66449432, 17454580, 211083730, 1171704435, 1955808460, 1171704435, 211083730, 2198596400, 16476937840, 40121261136, 40121261136, 16476937840, 2198596400, 20465052608, 196924458720, 647739636160
Offset: 6

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Author

Gheorghe Coserea, Mar 15 2016

Keywords

Comments

Row n contains n-5 terms.

Examples

			Triangle starts:
n\f  [1]          [2]          [3]          [4]          [5]
[6]  1485;
[7]  56628,       56628;
[8]  1169740,     2668750,     1169740;
[9]  17454580,    66449432,    66449432,    17454580;
[10] 211083730,   1171704435,  1955808460,  1171704435,  211083730;
[11] ...

Links

Crossrefs

Columns f=1-10 give: A288075 f=1, A288076 f=2, A288077 f=3, A288078 f=4, A288079 f=5, A288080 f=6, A288081 f=7, A288262 f=8, A288263 f=9, A288264 f=10.
Row sums give A104742 (column 3 of A269919).
Cf. A269921, A269922, A269924, A269925.

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, 3], {n, 6, 12}, {f, 1, n-5}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)
  • PARI
    N = 12; G = 3; 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))))

A288263 a(n) is the number of rooted maps with n edges and 9 faces on an orientable surface of genus 3.

Original entry on oeis.org

1384928666550, 176357530955320, 10933959720960760, 447708887118504600, 13767319160210071404, 341505418008822731328, 7151648337964982801760, 130468023103972196647776, 2121333601263313429701060, 31276917257222840819283888, 423834000658990977141751472, 5335660046838578422013157200
Offset: 14

Views

Author

Gheorghe Coserea, Jun 07 2017

Keywords

Links

Crossrefs

Rooted maps of genus 3 with n edges and f faces for 1<=f<=10: A288075 f=1, A288076 f=2, A288077 f=3, A288078 f=4, A288079 f=5, A288080 f=6, A288081 f=7, A288262 f=8, this sequence, A288264 f=10.
Column 9 of A269923.
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) ((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}]);
a[n_] := Q[n, 9, 3];
Table[a[n], {n, 14, 25}] (* Jean-François Alcover, Oct 17 2018 *)

A288264 a(n) is the number of rooted maps with n edges and 10 faces on an orientable surface of genus 3.

Original entry on oeis.org

10369994005800, 1461629029629340, 99727841192820016, 4470547991985864322, 149789855223187292608, 4031165546220945277040, 91230456810047671200128, 1792206112041706943912462, 31276917257222840819283888, 493477269339182312960416344, 7136207296287499744197970400, 95626920613336304647976494116
Offset: 15

Views

Author

Gheorghe Coserea, Jun 07 2017

Keywords

Links

Crossrefs

Rooted maps of genus 3 with n edges and f faces for 1<=f<=10: A288075 f=1, A288076 f=2, A288077 f=3, A288078 f=4, A288079 f=5, A288080 f=6, A288081 f=7, A288262 f=8, A288263 f=9, this sequence.
Column 10 of A269923.
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) ((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}]);
a[n_] := Q[n, 10, 3];
Table[a[n], {n, 15, 26}] (* Jean-François Alcover, Oct 17 2018 *)

A288076 a(n) is the number of rooted maps with n edges and 2 faces on an orientable surface of genus 3.

Original entry on oeis.org

56628, 2668750, 66449432, 1171704435, 16476937840, 196924458720, 2079913241120, 19925913354061, 176357530955320, 1461629029629340, 11460411934448048, 85694099173907510, 614960028331370816, 4257157940494918160, 28549761695867223680, 186131532080726321441, 1183191417356212860200, 7351865732351585503652
Offset: 7

Views

Author

Gheorghe Coserea, Jun 07 2017

Keywords

Links

Crossrefs

Rooted maps of genus 3 with n edges and f faces for 1<=f<=10: A288075 f=1, this sequence, A288077 f=3, A288078 f=4, A288079 f=5, A288080 f=6, A288081 f=7, A288262 f=8, A288263 f=9, A288264 f=10.
Column 2 of A269923.
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) ((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}]);
a[n_] := Q[n, 2, 3];
Table[a[n], {n, 7, 24}] (* Jean-François Alcover, Oct 17 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288076_ser(N) = {
  my(y = A000108_ser(N+1));
  y*(y-1)^7*(1485*y^6 + 111969*y^5 + 453295*y^4 - 389693*y^3 - 443894*y^2 + 361702*y - 38236)/(y-2)^20;
};
Vec(A288076_ser(18))

A288077 a(n) is the number of rooted maps with n edges and 3 faces on an orientable surface of genus 3.

Original entry on oeis.org

1169740, 66449432, 1955808460, 40121261136, 647739636160, 8789123742880, 104395235785256, 1115525500250760, 10933959720960760, 99727841192820016, 855779329367736840, 6968569097113244096, 54217755730994858080, 405300088876353160320, 2924455840981270327952, 20446207814548586119000, 138958722742591452843432
Offset: 8

Views

Author

Gheorghe Coserea, Jun 07 2017

Keywords

Links

Crossrefs

Rooted maps of genus 3 with n edges and f faces for 1<=f<=10: A288075 f=1, A288076 f=2, this sequence, A288078 f=4, A288079 f=5, A288080 f=6, A288081 f=7, A288262 f=8, A288263 f=9, A288264 f=10.
Column 3 of A269923.
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) ((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}]);
a[n_] := Q[n, 3, 3];
Table[a[n], {n, 8, 25}] (* Jean-François Alcover, Oct 17 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288077_ser(N) = {
  my(y = A000108_ser(N+1));
  -4*y*(y-1)^8*(28314*y^7 + 1229985*y^6 + 4821650*y^5 - 4914053*y^4 - 6967314*y^3 + 7429165*y^2 - 1071576*y - 263736)/(y-2)^23;
};
Vec(A288077_ser(17))

A288078 a(n) is the number of rooted maps with n edges and 4 faces on an orientable surface of genus 3.

Original entry on oeis.org

17454580, 1171704435, 40121261136, 945068384880, 17326957790896, 264477214235234, 3505018618003600, 41491242915292306, 447708887118504600, 4470547991985864322, 41790549086980226368, 369061676845849000520, 3101645444966543203008, 24954084939131951164980, 193145505023621965434976, 1444143475412182351017494, 10467259286591304015806600
Offset: 9

Views

Author

Gheorghe Coserea, Jun 07 2017

Keywords

Links

Crossrefs

Rooted maps of genus 3 with n edges and f faces for 1<=f<=10: A288075 f=1, A288076 f=2, A288077 f=3, this sequence, A288079 f=5, A288080 f=6, A288081 f=7, A288262 f=8, A288263 f=9, A288264 f=10.
Column 4 of A269923.
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) ((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}]);
a[n_] := Q[n, 4, 3];
Table[a[n], {n, 9, 26}] (* Jean-François Alcover, Oct 17 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288078_ser(N) = {
  my(y = A000108_ser(N+1));
  y*(y-1)^9*(5008230*y^8 + 164100330*y^7 + 620429875*y^6 - 742482075*y^5 - 1203385090*y^4 + 1546511666*y^3 - 224365292*y^2 - 189952744*y + 41589680)/(y-2)^26;
};
Vec(A288078_ser(17))

A288079 a(n) is the number of rooted maps with n edges and 5 faces on an orientable surface of genus 3.

Original entry on oeis.org

211083730, 16476937840, 647739636160, 17326957790896, 357391270819604, 6087558311398000, 89390908732820144, 1165172136542282424, 13767319160210071404, 149789855223187292608, 1518921342035154605600, 14492634832409091816640, 131114130730951689447016, 1131791523345860091265696, 9370402052804684247760928
Offset: 10

Views

Author

Gheorghe Coserea, Jun 07 2017

Keywords

Links

Crossrefs

Rooted maps of genus 3 with n edges and f faces for 1 <= f <= 10: A288075 (f = 1), A288076 (f = 2), A288077 (f = 3), A288078 (f = 4), this sequence (f = 5), A288080 (f = 6), A288081 (f = 7), A288262 (f = 8), A288263 (f = 9), A288264 (f = 10).
Column 5 of A269923.
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) ((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}]);
a[n_] := Q[n, 5, 3];
Table[a[n], {n, 10, 27}] (* Jean-François Alcover, Oct 17 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288079_ser(N) = {
  my(y = A000108_ser(N+1));
  -2*y*(y-1)^10*(83904012*y^9 + 2299548501*y^8 + 8375416306*y^7 - 11663434748*y^6 - 20521873396*y^5 + 30517603222*y^4 - 3781427784*y^3 - 7908127656*y^2 + 2862038656*y - 158105248)/(y-2)^29;
};
Vec(A288079_ser(15))

A288080 a(n) is the number of rooted maps with n edges and 6 faces on an orientable surface of genus 3.

Original entry on oeis.org

2198596400, 196924458720, 8789123742880, 264477214235234, 6087558311398000, 114899070275212424, 1857975645023518752, 26522236056202555206, 341505418008822731328, 4031165546220945277040, 44171448380277095027584, 453764845712090669861060, 4405234525240663358548000, 40682085269643556632419504, 359336179016097679450360000
Offset: 11

Views

Author

Gheorghe Coserea, Jun 07 2017

Keywords

Links

Crossrefs

Rooted maps of genus 3 with n edges and f faces for 1<=f<=10: A288075 f=1, A288076 f=2, A288077 f=3, A288078 f=4, A288079 f=5, this sequence, A288081 f=7, A288262 f=8, A288263 f=9, A288264 f=10.
Column 6 of A269923.
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) ((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}]);
a[n_] := Q[n, 6, 3];
Table[a[n], {n, 11, 28}] (* Jean-François Alcover, Oct 17 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288080_ser(N) = {
  my(y = A000108_ser(N+1));
  2*y*(y-1)^11*(2360692395*y^10 + 57065162931*y^9 + 200199438395*y^8 - 321653197109*y^7 - 594662939878*y^6 + 999754510326*y^5 - 90653073868*y^4 - 435707439920*y^3 + 201952082336*y^2 - 14180151168*y - 3375786240)/(y-2)^32;
};
Vec(A288080_ser(15))

A288081 a(n) is the number of rooted maps with n edges and 7 faces on an orientable surface of genus 3.

Original entry on oeis.org

20465052608, 2079913241120, 104395235785256, 3505018618003600, 89390908732820144, 1857975645023518752, 32904419378927915376, 511895831411154922176, 7151648337964982801760, 91230456810047671200128, 1076401288635137599528944, 11867194568934207062990560
Offset: 12

Views

Author

Gheorghe Coserea, Jun 07 2017

Keywords

Links

Crossrefs

Rooted maps of genus 3 with n edges and f faces for 1<=f<=10: A288075 f=1, A288076 f=2, A288077 f=3, A288078 f=4, A288079 f=5, A288080 f=6, this sequence, A288262 f=8, A288263 f=9, A288264 f=10.
Column 7 of A269923.
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, 7, 3];
Table[a[n], {n, 12, 27}] (* Jean-François Alcover, Oct 17 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288081_ser(N) = {
  my(y = A000108_ser(N+1));
  -8*y*(y-1)^12*(14699198844*y^11 + 323418619692*y^10 + 1093150970776*y^9 - 2010290018547*y^8 - 3822380209098*y^7 + 7160304314725*y^6 - 371305853280*y^5 - 4606441266688*y^4 + 2480182576832*y^3 - 129107145168*y^2 - 150618243904*y + 20945187392)/(y-2)^35;
};
Vec(A288081_ser(12))

A288262 a(n) is the number of rooted maps with n edges and 8 faces on an orientable surface of genus 3.

Original entry on oeis.org

174437377400, 19925913354061, 1115525500250760, 41491242915292306, 1165172136542282424, 26522236056202555206, 511895831411154922176, 8640883781524178188980, 130468023103972196647776, 1792206112041706943912462, 22695416350294243544684240, 267740228837597817351215676
Offset: 13

Views

Author

Gheorghe Coserea, Jun 07 2017

Keywords

Links

Crossrefs

Rooted maps of genus 3 with n edges and f faces for 1<=f<=10: A288075 f=1, A288076 f=2, A288077 f=3, A288078 f=4, A288079 f=5, A288080 f=6, A288081 f=7, this sequence, A288263 f=9, A288264 f=10.
Column 8 of A269923.
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, 8, 3];
Table[a[n], {n, 13, 24}] (* Jean-François Alcover, Oct 17 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288262_ser(N) = {
  my(y = A000108_ser(N+1));
  y*(y-1)^13*(2675326679856*y^12 + 54684388381464*y^11 + 178122315841075*y^10 - 372236561648447*y^9 - 717438005317146*y^8 + 1482970059363466*y^7 - 17264319660476*y^6 - 1294789702753096*y^5 + 770104389507952*y^4 - 4493523304288*y^3 - 105563098094272*y^2 + 24298454684800*y - 895286303488)/(y-2)^38;
};
Vec(A288262_ser(12))
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