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.

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

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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))))

A288276 a(n) is the number of rooted maps with n edges and 6 faces on an orientable surface of genus 4.

Original entry on oeis.org

1480593013900, 160576594766588, 8615949311310872, 309197871098871838, 8419549939292302908, 186553519919803261860, 3515647035511186627416, 58089920897558352891672, 860337164444236894357488, 11612741439751867739074432, 144715531380208437909370144, 1682205432436689960841795876
Offset: 13

Views

Author

Gheorghe Coserea, Jun 08 2017

Keywords

Links

Crossrefs

Rooted maps of genus 4 with n edges and f faces for 1<=f<=10: A288271 f=1, A288272 f=2, A288273 f=3, A288274 f=4, A288275 f=5, this sequence, A288277 f=7, A288278 f=8, A288279 f=9, A288280 f=10.
Column 6 of A269924.
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, 4];
Table[a[n], {n, 13, 24}] (* Jean-François Alcover, Oct 16 2018 *)

Formula

G.f.: 2*y*(y-1)^13*(1208305403982*y^12 + 42344287039512*y^11 + 283047148578040*y^10 + 47183718440672*y^9 - 1618438221531593*y^8 + 617910272368381*y^7 + 2488374601412831*y^6 - 2268379207704481*y^5 - 116197489174642*y^4 + 764144804102008*y^3 - 252877960850800*y^2 + 8651012216320*y + 3769026206720)/(y-2)^38, where y=A000108(x).

A288271 a(n) is the number of rooted maps with n edges and one face on an orientable surface of genus 4.

Original entry on oeis.org

225225, 12317877, 351683046, 7034538511, 111159740692, 1480593013900, 17302190625720, 182231849209410, 1763184571730010, 15894791312284170, 134951136993773100, 1088243826731751690, 8391311316938069520, 62210659883935683120, 445441857820701181440, 3092035882104030618900
Offset: 8

Views

Author

Gheorghe Coserea, Jun 08 2017

Keywords

Links

Crossrefs

Rooted maps of genus 4 with n edges and f faces for 1<=f<=10: this sequence, 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.
Column 1 of A269924.
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, 1, 4];
Table[a[n], {n, 8, 23}] (* Jean-François Alcover, Oct 16 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288271_ser(N) = {
  my(y = A000108_ser(N+1));
  -143*y*(y-1)^8*(1575*y^6 + 13689*y^5 + 4689*y^4 - 34417*y^3 + 11361*y^2 + 7017*y - 2339)/(y-2)^23;
};
Vec(A288271_ser(16))

Formula

G.f.: -143*y*(y-1)^8*(1575*y^6 + 13689*y^5 + 4689*y^4 - 34417*y^3 + 11361*y^2 + 7017*y - 2339)/(y-2)^23, where y=A000108(x).

A288272 a(n) is the number of rooted maps with n edges and 2 faces on an orientable surface of genus 4.

Original entry on oeis.org

12317877, 792534015, 26225260226, 600398249550, 10743797911132, 160576594766588, 2089035241981688, 24325590127655531, 258634264294653390, 2548272396065512974, 23532893106071038404, 205518653220527665304, 1709552077642556424368, 13623964536133602210560, 104522878918062035228512
Offset: 9

Views

Author

Gheorghe Coserea, Jun 08 2017

Keywords

Links

Crossrefs

Rooted maps of genus 4 with n edges and f faces for 1<=f<=10: A288271 f=1, this sequence, A288273 f=3, A288274 f=4, A288275 f=5, A288276 f=6, A288277 f=7, A288278 f=8, A288279 f=9, A288280 f=10.
Column 2 of A269924.
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, 4];
Table[a[n], {n, 9, 23}] (* Jean-François Alcover, Oct 16 2018 *)

Formula

G.f.: y*(y-1)^9*(225225*y^8 + 25467156*y^7 + 207300366*y^6 + 77853486*y^5 - 660073489*y^4 + 222312257*y^3 + 269246651*y^2 - 140048085*y + 10034310)/(y-2)^26, where y=A000108(x).

A288273 a(n) is the number of rooted maps with n edges and 3 faces on an orientable surface of genus 4.

Original entry on oeis.org

351683046, 26225260226, 993494827480, 25766235457300, 517592962672296, 8615949311310872, 123981042854132536, 1587135819804394530, 18451302662846918700, 197822824662547694148, 1979281881126113225376, 18654346303702719912848, 166862901890503876520320, 1425340713681247480547040, 11686190470805703242554960
Offset: 10

Views

Author

Gheorghe Coserea, Jun 08 2017

Keywords

Links

Crossrefs

Rooted maps of genus 4 with n edges and f faces for 1<=f<=10: A288271 f=1, A288272 f=2, this sequence, A288274 f=4, A288275 f=5, A288276 f=6, A288277 f=7, A288278 f=8, A288279 f=9, A288280 f=10.
Column 3 of A269924.
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, 4];
Table[a[n], {n, 10, 24}] (* Jean-François Alcover, Oct 16 2018 *)

Formula

G.f.: -2*y*(y-1)^10*(12317877*y^9 + 793781118*y^8 + 6094043038*y^7 + 2216299748*y^6 - 23375789497*y^5 + 7963356801*y^4 + 15368481377*y^3 - 10027219339*y^2 + 877859200*y + 252711200)/(y-2)^29, where y=A000108(x).

A288275 a(n) is the number of rooted maps with n edges and 5 faces on an orientable surface of genus 4.

Original entry on oeis.org

111159740692, 10743797911132, 517592962672296, 16789118602155860, 415691294404230748, 8419549939292302908, 145737674581607574840, 2221381417843144801098, 30468100266480917147760, 382217975972687580876304, 4441222132558609054169216, 48280421251792089554320464
Offset: 12

Views

Author

Gheorghe Coserea, Jun 08 2017

Keywords

Links

Crossrefs

Rooted maps of genus 4 with n edges and f faces for 1<=f<=10: A288271 f=1, A288272 f=2, A288273 f=3, A288274 f=4, this sequence, A288276 f=6, A288277 f=7, A288278 f=8, A288279 f=9, A288280 f=10.
Column 5 of A269924.
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, 4];
Table[a[n], {n, 12, 23}] (* Jean-François Alcover, Oct 16 2018 *)

Formula

G.f.: -2*y*(y-1)^12*(33259798737*y^11 + 1329990099093*y^10 + 9262655718313*y^9 + 2336641955449*y^8 - 47227883527259*y^7 + 17056753299711*y^6 + 58186472373731*y^5 - 48817840576153*y^4 + 819511081872*y^3 + 9462230411332*y^2 - 2475017890416*y + 88807125936)/(y-2)^35, where y=A000108(x).

A288277 a(n) is the number of rooted maps with n edges and 7 faces on an orientable surface of genus 4.

Original entry on oeis.org

17302190625720, 2089035241981688, 123981042854132536, 4892650539994184868, 145737674581607574840, 3515647035511186627416, 71823371612912533887168, 1281537868340178808063824, 20423544863369526066131328, 295680368360952875467454880, 3940377769373862621216994864
Offset: 14

Views

Author

Gheorghe Coserea, Jun 08 2017

Keywords

Links

Crossrefs

Rooted maps of genus 4 with n edges and f faces for 1<=f<=10: A288271 f=1, A288272 f=2, A288273 f=3, A288274 f=4, A288275 f=5, A288276 f=6, this sequence, A288278 f=8, A288279 f=9, A288280 f=10.
Column 7 of A269924.
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, 7, 4];
Table[a[n], {n, 14, 24}] (* Jean-François Alcover, Oct 16 2018 *)

Formula

G.f.: -4*y*(y-1)^14*(18995313191166*y^13 + 602583747147072*y^12 + 3880832501643076*y^11 + 259447266126966*y^10 - 24577880734142257*y^9 + 10075843752456953*y^8 + 45406701745704921*y^7 - 44360505974166179*y^6 - 5860774604042624*y^5 + 22759971294835512*y^4 - 8598423383057104*y^3 - 18688742922288*y^2 + 464831946526080*y - 48608581644864)/(y-2)^41, where y=A000108(x).

A288278 a(n) is the number of rooted maps with n edges and 8 faces on an orientable surface of genus 4.

Original entry on oeis.org

182231849209410, 24325590127655531, 1587135819804394530, 68503375296263488977, 2221381417843144801098, 58089920897558352891672, 1281537868340178808063824, 24605894500188479477960928, 420612140517667008915254376, 6512251870890866709301451550, 92559480623350598649493386580
Offset: 15

Views

Author

Gheorghe Coserea, Jun 08 2017

Keywords

Links

Crossrefs

Rooted maps of genus 4 with n edges and f faces for 1<=f<=10: A288271 f=1, A288272 f=2, A288273 f=3, A288274 f=4, A288275 f=5, A288276 f=6, A288277 f=7, this sequence, A288279 f=9, A288280 f=10.
Column 8 of A269924.
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, 8, 4];
Table[a[n], {n, 15, 25}] (* Jean-François Alcover, Oct 16 2018 *)

Formula

G.f.: y*(y-1)^15*(2141204115631518*y^14 + 62802256981978404*y^13 + 390904315702808387*y^12 - 17469926941849537*y^11 - 2715522908192830943*y^10 + 1209526054185992549*y^9 + 5862111891800632315*y^8 - 6084780630540788053*y^7 - 1344178041537337418*y^6 + 4359417524034703460*y^5 - 1779344954166712472*y^4 - 128701285301543888*y^3 + 220665627694548576*y^2 - 38233669153240512*y + 844773167217024)/(y-2)^44, where y=A000108(x).

A288279 a(n) is the number of rooted maps with n edges and 9 faces on an orientable surface of genus 4.

Original entry on oeis.org

1763184571730010, 258634264294653390, 18451302662846918700, 866831237081712285138, 30468100266480917147760, 860337164444236894357488, 20423544863369526066131328, 420612140517667008915254376, 7689357064107454375292572788, 126977551039680427095997314540, 1920060399356995304343259728312
Offset: 16

Views

Author

Gheorghe Coserea, Jun 08 2017

Keywords

Links

Crossrefs

Rooted maps of genus 4 with n edges and f faces for 1<=f<=10: A288271 f=1, A288272 f=2, A288273 f=3, A288274 f=4, A288275 f=5, A288276 f=6, A288277 f=7, A288278 f=8, this sequence, A288280 f=10.
Column 9 of A269924.
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, 4];
Table[a[n], {n, 16, 26}] (* Jean-François Alcover, Oct 16 2018 *)

Formula

G.f.: -6*y*(y-1)^16*(9225445001552610*y^15 + 253889174613116085*y^14 + 1531144661703557241*y^13 - 254390452688914375*y^12 - 11576322921612113581*y^11 + 5646113444605154169*y^10 + 28587502564009313669*y^9 - 31350769849259642447*y^8 - 9832935993984430480*y^7 + 29500732589692418132*y^6 - 12567984363713561312*y^5 - 2218978200544343392*y^4 + 2888444088307833216*y^3 - 630076702195212352*y^2 + 8436883230156800*y + 6263496930404352)/(y-2)^47, where y=A000108(x).

A288280 a(n) is the number of rooted maps with n edges and 10 faces on an orientable surface of genus 4.

Original entry on oeis.org

15894791312284170, 2548272396065512974, 197822824662547694148, 10071757699155275906824, 382217975972687580876304, 11612741439751867739074432, 295680368360952875467454880, 6512251870890866709301451550, 126977551039680427095997314540, 2230836871835420574103711453068
Offset: 17

Views

Author

Gheorghe Coserea, Jun 08 2017

Keywords

Links

Crossrefs

Rooted maps of genus 4 with n edges and f faces for 1<=f<=10: 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, this sequence.
Column 10 of A269924.
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) (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] (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, 4];
Table[a[n], {n, 17, 26}] (* Jean-François Alcover, Oct 16 2018 *)

Formula

G.f.: 2*y*(y-1)^17*(667113335854505289*y^16 + 17412039201241985652*y^15 + 101949739105950626070*y^14 - 30202970169901595562*y^13 - 833532476362240891879*y^12 + 447114036864981439647*y^11 + 2316066844919602997013*y^10 - 2673632819222127570107*y^9 - 1088786810085394834566*y^8 + 3157924186313124711792*y^7 - 1371258409341666011952*y^6 - 433458368694714259536*y^5 + 515333809963509426144*y^4 - 126279314363368987008*y^3 - 3637814234318456832*y^2 + 4694513255143047936*y - 365353090019990016)/(y-2)^50, where y=A000108(x).
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