A288073 -id:A288073 - cp's OEIS frontend

A269921 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 1.

1, 10, 10, 70, 167, 70, 420, 1720, 1720, 420, 2310, 14065, 24164, 14065, 2310, 12012, 100156, 256116, 256116, 100156, 12012, 60060, 649950, 2278660, 3392843, 2278660, 649950, 60060, 291720, 3944928, 17970784, 36703824, 36703824, 17970784

Offset: 2

Gheorghe Coserea, Mar 14 2016

Row n contains n-1 terms.

			Triangle starts:
n\f    [1]      [2]      [3]      [4]      [5]      [6]      [7]
[2]    1;
[3]    10,      10;
[4]    70,      167,     70;
[5]    420,     1720,    1720,    420;
[6]    2310,    14065,   24164,   14065,   2310;
[7]    12012,   100156,  256116,  256116,  100156,  12012;
[8]    60060,   649950,  2278660, 3392843, 2278660, 649950,  60060;
[9]    ...

Gheorghe Coserea, Rows n = 2..202, flattened
Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

Columns f=1-10 give: A002802 f=1, A006295 f=2, A006296 f=3, A288071 f=4, A288072 f=5, A287046 f=6, A287047 f=7, A287048 f=8, A288073 f=9, A288074 f=10.
Row sums give A006300 (column 1 of A269919).
Cf. A006297 (row maxima).

M = 9; G = 1; gMax[n_] := Min[Quotient[n, 2], G];
Q = Array[0&, {M + 1, M + 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_] := (Qset[0, 0, x]; For[n = 1, n <= Length[Q] - 1, n++, For[g = 0, g <= gMax[n], g++, 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[ Sum[(2*k - 1) * (2*(n - k) - 1) * Qget[k - 1, i] * Qget[n - k - 1, g - i], {i, 0, g}], {k, 1, n-1}]; Qset[n, g, (t1 + t2 + t3) * 6/(n+1)]]]);
Quadric[x];
(List @@@ Table[Qget[n - 1 + 2*G, G] // Expand, {n, 1, M + 1 - 2*G}]) /. x -> 1 // Flatten (* Jean-François Alcover, Jun 13 2017, adapted from PARI *)

N = 9; G = 1; 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))))

A006295 Number of genus 1 rooted maps with 2 faces with n vertices.

Original entry on oeis.org

10, 167, 1720, 14065, 100156, 649950, 3944928, 22764165, 126264820, 678405090, 3550829360, 18182708362, 91392185080, 452077562620, 2205359390592, 10627956019245, 50668344988068, 239250231713210, 1120028580999440, 5202779260636958, 23998704563581000, 109991785264412452

Offset: 3

N. J. A. Sloane

nonn

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.

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
W. T. Tutte, On the enumeration of planar maps, Bull. Amer. Math. Soc. 74 1968 64-74.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
Notes

Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, this sequence, A006296 f=3, A288071 f=4, A288072 f=5, A287046 f=6, A287047 f=7, A287048 f=8, A288073 f=9, A288074 f=10.

Column 2 of A269921, column 1 of A270406.

Rest[CoefficientList[Series[(1 - Sqrt[1 - 4 x]) (11 + 12 x + 9 Sqrt[1 - 4 x]) / (4 (4 x - 1)^4), {x, 0, 40}], x]] (* Vincenzo Librandi, Jun 06 2017 *)

A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A006295_ser(N) = {
  my(y = A000108_ser(N+1));  y*(y-1)^3*(y^2 + 15*y - 6)/(y-2)^8;
};
Vec(A006295_ser(31)) \\ Gheorghe Coserea, Jun 04 2017

my(x = 'x + O('x^60)); Vec(x*(1-sqrt(1-4*x))*(11+12*x+9*sqrt(1-4*x))/(4*(4*x-1)^4)) \\ Michel Marcus, Jun 05 2017

G.f.: x*(1-sqrt(1-4*x))*(11+12*x+9*sqrt(1-4*x))/(4*(4*x-1)^4). - Sean A. Irvine, Nov 14 2010

More terms from Sean A. Irvine, Nov 14 2010

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

N. J. A. Sloane

nonn

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.

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
Notes

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.

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 *)

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

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

More terms from Sean A. Irvine, Nov 14 2010

A287046 a(n) is the number of rooted maps with n edges and 6 faces on an orientable surface of genus 1.

Original entry on oeis.org

12012, 649950, 17970784, 344468530, 5188948072, 65723863196, 729734918432, 7302676928666, 67173739068760, 576218752277476, 4660202610532480, 35839052357422132, 263868150558327376, 1870153808268516280, 12816868756802256832, 85256107136168684650, 552171259884681058744

Offset: 7

Gheorghe Coserea, Jun 04 2017

nonn

Sean R. Carrell and Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.
Sean R. Carrell and Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, Journal of Combinatorial Theory, Series A, 133 (2015), 58-75.

Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, A006295 f=2, A006296 f=3, A288071 f=4, A288072 f=5, this sequence, A287047 f=7, A287048 f=8, A288073 f=9, A288074 f=10.
Column 6 of A269921, column 1 of A270410.
Cf. A000108.

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, 6, 1];
Table[a[n], {n, 7, 23}] (* Jean-François Alcover, Oct 17 2018 *)

A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A287046_ser(N) = {
  my(y = A000108_ser(N+1));
  2*y*(y-1)^7*(28457*y^6 + 179171*y^5 - 222214*y^4 - 172512*y^3 + 257232*y^2 - 59904*y - 4224)/(y-2)^20;
};
Vec(A287046_ser(17))

A287047 a(n) is the number of rooted maps with n edges and 7 faces on an orientable surface of genus 1.

Original entry on oeis.org

60060, 3944928, 129726760, 2908358552, 50534154408, 729734918432, 9145847808784, 102432266545800, 1046677747672360, 9908748651241088, 87930943305742512, 738178726378902064, 5905479331377981200, 45289976937922983360, 334600965220354244896, 2391127223524518889064, 16585285393291515557928

Offset: 8

Gheorghe Coserea, Jun 04 2017

nonn

Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, A006295 f=2, A006296 f=3, A288071 f=4, A288072 f=5, A287046 f=6, this sequence, A287048 f=8, A288073 f=9, A288074 f=10.
Column 7 of A269921, column 1 of A270411.
Cf. A000108.

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, 1];
Table[a[n], {n, 8, 24}] (* Jean-François Alcover, Oct 18 2018 *)

A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A287047_ser(N) = {
  my(y = A000108_ser(N+1));
  -4*y*(y-1)^8*(184142*y^7 + 1083793*y^6 - 1540136*y^5 - 1481152*y^4 + 2626176*y^3 - 737232*y^2 - 184896*y + 64320)/(y-2)^23;
};
Vec(A287047_ser(17))

A287048 a(n) is the number of rooted maps with n edges and 8 faces on an orientable surface of genus 1.

Original entry on oeis.org

291720, 22764165, 875029804, 22620890127, 448035881592, 7302676928666, 102432266545800, 1274461449989715, 14373136466094880, 149314477245194262, 1446563778096423816, 13196809961724011350, 114253624700659216080, 944690705838217837620, 7498935691376059259344, 57398464959432306918747

Offset: 9

Gheorghe Coserea, Jun 04 2017

nonn

Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, A006295 f=2, A006296 f=3, A288071 f=4, A288072 f=5, A287046 f=6, A287047 f=7, this sequence, A288073 f=9, A288074 f=10.
Column 8 of A269921; column 1 of A270412.
Cf. A000108.

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, 1];
Table[a[n], {n, 9, 25}] (* Jean-François Alcover, Oct 18 2018 *)

A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A287048_ser(N) = {
  my(y = A000108_ser(N+1));
  y*(y-1)^9*(9370183*y^8 + 52321971*y^7 - 83853806*y^6 - 93946092*y^5 + 189910936*y^4 - 57493776*y^3 - 31383264*y^2 + 16878912*y - 1513344)/(y-2)^26;
};
Vec(A287048_ser(16))

A288071 a(n) is the number of rooted maps with n edges and 4 faces on an orientable surface of genus 1.

Original entry on oeis.org

420, 14065, 256116, 3392843, 36703824, 344468530, 2908358552, 22620890127, 164767964504, 1137369687454, 7506901051000, 47700234551918, 293370096957504, 1753945289216484, 10229201477344752, 58364244137596695, 326571194881454376, 1795631576981016038, 9718877491130813368, 51858415558095569962

Offset: 5

Gheorghe Coserea, Jun 05 2017

nonn

Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, A006295 f=2, A006296 f=3, this sequence, A288072 f=5, A287046 f=6, A287047 f=7, A287048 f=8, A288073 f=9, A288074 f=10.
Column 4 of A269921.
Cf. A000108.

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, 1];
Table[a[n], {n, 5, 24}] (* Jean-François Alcover, Oct 18 2018 *)

A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288071_ser(N) = {
  my(y = A000108_ser(N+1));
  y*(y-1)^5*(307*y^4 + 2411*y^3 - 2094*y^2 - 708*y + 504)/(y-2)^14;
};
Vec(A288071_ser(20))

A288072 a(n) is the number of rooted maps with n edges and 5 faces on an orientable surface of genus 1.

Original entry on oeis.org

2310, 100156, 2278660, 36703824, 472592916, 5188948072, 50534154408, 448035881592, 3682811916980, 28442316247080, 208462422428152, 1461307573813824, 9857665477085832, 64309102366765200, 407372683115470800, 2514120288996270024, 15159074541052024308, 89512241718624419624

Offset: 6

Gheorghe Coserea, Jun 05 2017

nonn

Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, A006295 f=2, A006296 f=3, A288071 f=4, this sequence, A287046 f=6, A287047 f=7, A287048 f=8, A288073 f=9, A288074 f=10.
Column 5 of A269921.
Cf. A000108.

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, 1];
Table[a[n], {n, 6, 23}] (* Jean-François Alcover, Oct 18 2018 *)

A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288072_ser(N) = {
  my(y = A000108_ser(N+1));
  -2*y*(y-1)^6*(2140*y^5 + 14751*y^4 - 15604*y^3 - 8820*y^2 + 10176*y - 1488)/(y-2)^17;
};
Vec(A288072_ser(18))

A288074 a(n) is the number of rooted maps with n edges and 10 faces on an orientable surface of genus 1.

Original entry on oeis.org

6466460, 678405090, 34225196720, 1137369687454, 28442316247080, 576218752277476, 9908748651241088, 149314477245194262, 2017523504473479992, 24868664942648145372, 283389619978690157408, 3017066587822315930220, 30265092793614787511376, 288055728071446557904968, 2616366012933033221518720

Offset: 11

Gheorghe Coserea, Jun 05 2017

nonn

Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, A006295 f=2, A006296 f=3, A288071 f=4, A288072 f=5, A287046 f=6, A287047 f=7, A287048 f=8, A288073 f=9, this sequence.

Column 10 of A269921.

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, 1];
Table[a[n], {n, 11, 25}] (* Jean-François Alcover, Oct 18 2018 *)

A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288074_ser(N) = {
  my(y = A000108_ser(N+1));
  2*y*(y-1)^11*(734641583*y^10 + 3795452665*y^9 - 7483071778*y^8 - 10235465624*y^7 + 25178445968*y^6 - 7563355856*y^5 - 11624244832*y^4 + 8854962048*y^3 - 1433163264*y^2 - 286758144*y + 65790464)/(y-2)^32;
};
Vec(A288074_ser(15))

cp's OEIS Frontend

A269921 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 1.

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A006295 Number of genus 1 rooted maps with 2 faces with n vertices.

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

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A287046 a(n) is the number of rooted maps with n edges and 6 faces on an orientable surface of genus 1.

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A287047 a(n) is the number of rooted maps with n edges and 7 faces on an orientable surface of genus 1.

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A287048 a(n) is the number of rooted maps with n edges and 8 faces on an orientable surface of genus 1.

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A288071 a(n) is the number of rooted maps with n edges and 4 faces on an orientable surface of genus 1.

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A288072 a(n) is the number of rooted maps with n edges and 5 faces on an orientable surface of genus 1.

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