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 1-9 of 9 results.

A269922 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 2.

Original entry on oeis.org

21, 483, 483, 6468, 15018, 6468, 66066, 258972, 258972, 66066, 570570, 3288327, 5554188, 3288327, 570570, 4390386, 34374186, 85421118, 85421118, 34374186, 4390386, 31039008, 313530000, 1059255456, 1558792200, 1059255456, 313530000, 31039008
Offset: 4

Views

Author

Gheorghe Coserea, Mar 15 2016

Keywords

Comments

Row n contains n-3 terms.

Examples

			Triangle starts:
n\f  [1]        [2]        [3]        [4]        [5]        [6]
[4]  21;
[5]  483,       483;
[6]  6468,      15018,     6468;
[7]  66066,     258972,    258972,    66066;
[8]  570570,    3288327,   5554188,   3288327,   570570;
[9]  4390386,   34374186,  85421118,  85421118,  34374186,  4390386;
[10] ...

Links

Crossrefs

Columns f=1-10 give: A006298 f=1, A288082 f=2, A288083 f=3, A288084 f=4, A288085 f=5, A288086 f=6, A288087 f=7, A288088 f=8, A288089 f=9, A288090 f=10.
Row sums give A006301 (column 2 of A269919).
Cf. A006299 (row maxima), A269921.

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, 2], {n, 4, 10}, {f, 1, n-3}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)
  • PARI
    N = 10; G = 2; 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))))

A288082 a(n) is the number of rooted maps with n edges and 2 faces on an orientable surface of genus 2.

Original entry on oeis.org

483, 15018, 258972, 3288327, 34374186, 313530000, 2583699888, 19678611645, 140725699686, 955708437684, 6216591472728, 38985279745230, 236923660397172, 1401097546161936, 8089830217844928, 45732525474843801, 253705943922820830, 1383896652090932364, 7434748752218650632, 39394773780853063650
Offset: 5

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, this sequence, A288083 f=3, A288084 f=4, A288085 f=5, A288086 f=6, A288087 f=7, A288088 f=8, A288089 f=9, A288090 f=10.
Column 2 of A269922, column 2 of A270406.
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, 2, 2];
Table[a[n], {n, 5, 24}] (* Jean-François Alcover, Oct 18 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288082_ser(N) = {
  my(y = A000108_ser(N+1));
  3*y*(y-1)^5*(7*y^4 + 294*y^3 + 309*y^2 - 547*y + 98)/(y-2)^14;
};
Vec(A288082_ser(20))

A288083 a(n) is the number of rooted maps with n edges and 3 faces on an orientable surface of genus 2.

Original entry on oeis.org

6468, 258972, 5554188, 85421118, 1059255456, 11270290416, 106853266632, 925572602058, 7454157823560, 56532447160536, 407653880116680, 2815913391715452, 18743188498056288, 120789163612555200, 756589971284883792, 4621041111902656770, 27595482540212519064, 161490751719681569736
Offset: 6

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, this sequence, A288084 f=4, A288085 f=5, A288086 f=6, A288087 f=7, A288088 f=8, A288089 f=9, A288090 f=10.
Column 3 of A269922, column 2 of A270407.
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, 3, 2];
Table[a[n], {n, 6, 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);
A288083_ser(N) = {
  my(y = A000108_ser(N+1));
  -6*y*(y-1)^6*(161*y^5 + 4005*y^4 + 4173*y^3 - 10701*y^2 + 2880*y + 560)/(y-2)^17;
};
Vec(A288083_ser(18))

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))

A288085 a(n) is the number of rooted maps with n edges and 5 faces on an orientable surface of genus 2.

Original entry on oeis.org

570570, 34374186, 1059255456, 22555934280, 375708427812, 5235847653036, 63648856688592, 694146691745820, 6928413234959820, 64232028100704156, 559373367462490656, 4616545437250956192, 36362952155187558600, 274925536462366808760, 2004633652255211204832, 14152391716870383219492
Offset: 8

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, A288084 f=4, this sequence, A288086 f=6, A288087 f=7, A288088 f=8, A288089 f=9, A288090 f=10.
Column 5 of A269922, column 2 of A270409.
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, 5, 2];
Table[a[n], {n, 8, 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);
A288085_ser(N) = {
  my(y = A000108_ser(N+1));
  -6*y*(y-1)^8*(108346*y^7 + 1760421*y^6 + 1641979*y^5 - 7296839*y^4 + 2560152*y^3 + 2713196*y^2 - 1525104*y + 132944)/(y-2)^23;
};
Vec(A288085_ser(16))

A288087 a(n) is the number of rooted maps with n edges and 7 faces on an orientable surface of genus 2.

Original entry on oeis.org

31039008, 2583699888, 106853266632, 2979641557620, 63648856688592, 1117259292848016, 16842445235560944, 224686278407291148, 2710382626755160416, 30044423965980553536, 309859885439753598768, 3002524783711124880936, 27551689577648333614176, 240961534103705377359840, 2019318707410947848445792
Offset: 10

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, A288084 f=4, A288085 f=5, A288086 f=6, this sequence, A288088 f=8, A288089 f=9, A288090 f=10.
Column 7 of A269922, column 2 of A270411.
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, 2];
Table[a[n], {n, 10, 24}] (* Jean-François Alcover, Oct 18 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288087_ser(N) = {
  my(y = A000108_ser(N+1));
  -12*y*(y-1)^10*(20697615*y^9 + 275716321*y^8 + 211910021*y^7 - 1514443109*y^6 + 601694224*y^5 + 1328709592*y^4 - 1136750032*y^3 + 153705072*y^2 + 76788992*y - 15442112)/(y-2)^29;
};
Vec(A288087_ser(15))

A288088 a(n) is the number of rooted maps with n edges and 8 faces on an orientable surface of genus 2.

Original entry on oeis.org

205633428, 19678611645, 925572602058, 29079129795702, 694146691745820, 13518984452463630, 224686278407291148, 3286157560248860532, 43241609165618454096, 520516978029736518606, 5805858136761540452700, 60619447491266688750132, 597358002436877437320936, 5593151345725345725640044
Offset: 11

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, A288084 f=4, A288085 f=5, A288086 f=6, A288087 f=7, this sequence, A288089 f=9, A288090 f=10.
Column 8 of A269922, column 2 of A270412.
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, 2];
Table[a[n], {n, 11, 24}] (* Jean-François Alcover, Oct 18 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288088_ser(N) = {
  my(y = A000108_ser(N+1));
  3*y*(y-1)^11*(1455480376*y^10 + 18151410348*y^9 + 12284790745*y^8 - 111454641175*y^7 + 46880062914*y^6 + 129967691724*y^5 - 125047028168*y^4 + 14650142480*y^3 + 19075464224*y^2 - 6255822912*y + 360993920)/(y-2)^32;
};
Vec(A288088_ser(14))

A288089 a(n) is the number of rooted maps with n edges and 9 faces on an orientable surface of genus 2.

Original entry on oeis.org

1293938646, 140725699686, 7454157823560, 261637840342860, 6928413234959820, 148755268498286436, 2710382626755160416, 43241609165618454096, 617910462111714896820, 8044640800289827566756, 96690983139765469347024, 1084226645505246141589704, 11439196912435362172792536, 114351801899024314438876200
Offset: 12

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, A288084 f=4, A288085 f=5, A288086 f=6, A288087 f=7, A288088 f=8, this sequence, A288090 f=10.
Column 9 of A269922.
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, 9, 2];
Table[a[n], {n, 12, 25}] (* Jean-François Alcover, Oct 18 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288089_ser(N) = {
  my(y = A000108_ser(N+1));
  -6*y*(y-1)^12*(12205186004*y^11 + 144345246789*y^10 + 83883548039*y^9 - 978172313331*y^8 + 436600889944*y^7 + 1435650005364*y^6 - 1511798886368*y^5 + 121539026592*y^4 + 411304907520*y^3 - 171035694144*y^2 + 14120686592*y + 1573053440)/(y-2)^35;
};
Vec(A288089_ser(13))

A288090 a(n) is the number of rooted maps with n edges and 10 faces on an orientable surface of genus 2.

Original entry on oeis.org

7808250450, 955708437684, 56532447160536, 2200626948631386, 64232028100704156, 1511718920778951024, 30044423965980553536, 520516978029736518606, 8044640800289827566756, 112860842135424498808968, 1456882832375987896763184, 17491588653334242501297012, 197038603477850885815215480
Offset: 13

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, A288084 f=4, A288085 f=5, A288086 f=6, A288087 f=7, A288088 f=8, A288089 f=9, this sequence.
Column 10 of A269922.
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, 10, 2];
Table[a[n], {n, 13, 25}] (* Jean-François Alcover, Oct 18 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288090_ser(N) = {
  my(y = A000108_ser(N+1));
  6*y*(y-1)^13*(197300616213*y^12 + 2233379349250*y^11 + 1077980722075*y^10 - 16537713992125*y^9 + 7856375825902*y^8 + 29387232350368*y^7 - 33290642716432*y^6 + 994024496848*y^5 + 14078465181600*y^4 - 6737013421440*y^3 + 532103069696*y^2 + 244607984896*y - 34798091776)/(y-2)^38;
};
Vec(A288090_ser(13))
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