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))))

A006296 Number of genus 1 rooted maps with 3 faces with n vertices.

Original entry on oeis.org

70, 1720, 24164, 256116, 2278660, 17970784, 129726760, 875029804, 5593305476, 34225196720, 201976335288, 1156128848680, 6447533938280, 35155923872640, 187959014565840, 987658610225052, 5110652802256260, 26084524995672080, 131501187454625560, 655590388845975000, 3235463376771463288, 15820770680078552000, 76708503479715247920, 369046200766330733880, 1762793459781859039080, 8364468224596427692896, 39445646133672676352560, 184956513528952419546448, 862615498961026097997392, 4003067488703222112053760, 18489846573354278755829152, 85028133934182275077421180, 389398354121840111751946628, 1776360539933013004774353872, 8073622060225813990245976280, 36567311475673299914222851832
Offset: 4

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).
  • T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

Links

Crossrefs

Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, A006295 f=2, this sequence, A288071 f=4, A288072 f=5, A287046 f=6, A287047 f=7, A287048 f=8, A288073 f=9, A288074 f=10.
Column 3 of A269921, column g=1 of A270407.

Programs

  • Mathematica
    Rest[CoefficientList[Series[(1 - Sqrt[1 - 4 x]) (45 + 152 x + (25 + 8 x) Sqrt[1 - 4 x]) / (2 (1 - 4 x)^(11 / 2)), {x, 0, 40}], x]] (* Vincenzo Librandi, Jun 06 2017 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A006296_ser(N) = {
  my(y = A000108_ser(N+1));
  -2*y*(y-1)^4*(10*y^3 + 97*y^2 - 64*y - 8)/(y-2)^11;
};
Vec(A006296_ser(36)) \\ Gheorghe Coserea, Jun 04 2017

Formula

G.f.: x(1-sqrt(1-4*x))(45+152*x+(25+8*x)sqrt(1-4*x))/(2(1-4*x)^(11/2)). - Sean A. Irvine, Nov 14 2010

Extensions

More terms from Sean A. Irvine, Nov 14 2010

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))

A000184 Number of genus 0 rooted maps with 3 faces with n vertices.

Original entry on oeis.org

2, 22, 164, 1030, 5868, 31388, 160648, 795846, 3845020, 18211380, 84876152, 390331292, 1775032504, 7995075960, 35715205136, 158401506118, 698102372988, 3059470021316, 13341467466520, 57918065919924, 250419305769512, 1078769490401032, 4631680461623664, 19825379450255900, 84622558822506328, 360270317908904328, 1530148541536781488, 6484511936352543096, 27423786092731382000, 115756362341775227888
Offset: 2

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 3 of A269920.
Column 0 of A270407.
Cf. A000108, A029887.

Programs

  • Magma
    [n*((n+1)*(n+2)*Catalan(n+1) - 3*4^n)/12: n in [2..30]]; // G. C. Greubel, Jul 18 2024
  • Mathematica
    a[n_] := 1/12*(2^(n+1)*(2*n+1)!!/(n-1)!-3*4^n*n); Table[a[n], {n, 2, 31}] (* Jean-François Alcover, Mar 12 2014 *)
  • SageMath
    [n*(2*(2*n+1)*binomial(2*n,n) - 3*4^n)//12 for n in range(2,30)] # G. C. Greubel, Jul 18 2024

Formula

a(n) = 2 * A029887(n-2). - Ralf Stephan, Aug 17 2004
a(n) = 4^n*Gamma(n+3/2)/(3*sqrt(Pi)*Gamma(n)) - n*4^(n-1). - Mark van Hoeij, Jul 06 2010
From G. C. Greubel, Jul 18 2024: (Start)
a(n) = (n/12)*( (n+1)*(n+2)*Catalan(n+1) - 3*4^n ).
G.f.: x*(1 - sqrt(1 - 4*x))/(1-4*x)^(5/2).
E.g.f.: (x/3)*exp(2*x)*( - 3*exp(2*x) + 3*(1+2*x)*BesselI(0, 2*x) + (3+8*x)*BesselI(1, 2*x) + 2*x*BesselI(2, 2*x) ). (End)

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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