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.
Original entry on oeis.org
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
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] ...
-
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))))
A215402
Number of rooted maps of (orientable) genus 4 containing n edges.
Original entry on oeis.org
225225, 24635754, 1495900107, 66519597474, 2416610807964, 75981252764664, 2141204115631518, 55352670009315660, 1334226671709010578, 30347730709395639732, 657304672067357799042, 13652607304062788395788, 273469313030628783700080, 5306599156694095573465824, 100128328831437989131706976, 1842794650155970906232185656
Offset: 8
- Andrew Howroyd, Table of n, a(n) for n = 8..500
- Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).
- Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
- Alexander Mednykh, Alain Giorgetti, Enumeration of genus four maps by number of edges, Ars Mathematica Contemporanea 4 (2011), 351--361.
Rooted maps with n edges of genus g for 0 <= g <= 10:
A000168,
A006300,
A006301,
A104742, this sequence,
A238355,
A238356,
A238357,
A238358,
A238359,
A238360.
-
T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 4];
Table[a[n], {n, 8, 30}] (* Jean-François Alcover, Jul 20 2018 *)
-
A005159_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-12*x))/(6*x);
A215402_ser(N) = {
my(y=A005159_ser(N+1));
-y*(y-1)^8*(15812*y^12 - 189744*y^11 + 4708549*y^10 - 24892936*y^9 + 173908449*y^8 - 567987942*y^7 + 1743939189*y^6 - 3485359548*y^5 + 5448471852*y^4 - 6051484928*y^3 + 4633500336*y^2 - 2228416192*y + 517976128)/(81*(y-2)^17*(y+2)^10);
};
Vec(A215402_ser(16)) \\ Gheorghe Coserea, Jun 02 2017
A238355
Number of rooted maps of genus 5 containing n edges.
Original entry on oeis.org
59520825, 8608033980, 672868675017, 37680386599440, 1692352190653740, 64755027944420400, 2190839204960030106, 67194704604610557072, 1901727022434216910002, 50322107898515282999256, 1257582616997225194094310, 29916524874047762719113408, 681758763997451748190036272, 14960113428664295584816860864
Offset: 10
Rooted maps with n edges of genus g for 0 <= g <= 10:
A000168,
A006300,
A006301,
A104742,
A215402, this sequence,
A238356,
A238357,
A238358,
A238359,
A238360.
-
T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 5];
Table[a[n], {n, 10, 30}] (* Jean-François Alcover, Jul 20 2018 *)
-
\\ see A238396
-
A005159_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-12*x))/(6*x);
A238355_ser(N) = {
my(y=A005159_ser(N+1));
y*(y-1)^10*(3149956*y^16 - 50399296*y^15 + 1641189689*y^14 - 12178227918*y^13 + 118643174857*y^12 - 572499071300*y^11 + 2690451915197*y^10 - 8657342508522*y^9 + 23652302179098*y^8 - 49891059998872*y^7 + 84432024838000*y^6 - 112355956173344*y^5 + 115338024848256*y^4 - 88846084908160*y^3 + 48488699816960*y^2 - 16837415717888*y + 2841312026112)/(243*(y-2)^22*(y+2)^13);
};
Vec(A238355_ser(14)) \\ Gheorghe Coserea, Jun 02 2017
A238356
Number of rooted maps of genus 6 containing n edges.
Original entry on oeis.org
24325703325, 4416286056750, 425671555397220, 28948474436455224, 1558252224413413380, 70639804918689629112, 2802850363447807024080, 99911395098598706576856, 3259947795252652107008514, 98729808377337068918681196, 2805432194025270702468165744
Offset: 12
Rooted maps with n edges of genus g for 0 <= g <= 10:
A000168,
A006300,
A006301,
A104742,
A215402,
A238355, this sequence,
A238357,
A238358,
A238359,
A238360.
-
T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 6];
Table[a[n], {n, 12, 30}] (* Jean-François Alcover, Jul 20 2018 *)
-
\\ see A238396
-
A005159_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-12*x))/(6*x);
A238356_ser(N) = {
my(y=A005159_ser(N+1));
-y*(y-1)^12*(3091382412*y^20 - 61827648240*y^19 + 2494741456179*y^18 - 23821030780564*y^17 + 297709107215018*y^16 - 1898397937026724*y^15 + 11996625283021532*y^14 - 53079600835119544*y^13 + 206468965657569764*y^12 - 637634273350412392*y^11 + 1660605297373850222*y^10 - 3573247507645221112*y^9 + 6390852378647917144*y^8 - 9449999309170921856*y^7 + 11435897504002339264*y^6 - 11175919884930946304*y^5 + 8621441033651120896*y^4 - 5068129528843341824*y^3 + 2141653827725309440*y^2 - 581932716954417152*y + 76958488611567616)/(2187*(y-2)^27*(y+2)^16);
};
Vec(A238356_ser(11)) \\ Gheorghe Coserea, Jun 02 2017
A238357
Number of genus-7 rooted maps with n edges.
Original entry on oeis.org
14230536445125, 3128879373858000, 360626952084151500, 29001816720933903504, 1828003659229082834100, 96187365300257285300064, 4395215998078319892167640, 179153431308203084149883760, 6641365771586560905099092466, 227189907562197156785567456832, 7252879937219595844346639732688
Offset: 14
Rooted maps with n edges of genus g for 0 <= g <= 10:
A000168,
A006300,
A006301,
A104742,
A215402,
A238355,
A238356, this sequence,
A238358,
A238359,
A238360.
-
T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 7];
Table[a[n], {n, 14, 30}] (* Jean-François Alcover, Jul 20 2018 *)
-
\\ see A238396
-
system("wget http://oeis.org/A238357/a238357.txt");
A005159_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-12*x))/(6*x);
A238357_ser(N) = subst(read("a238357.txt"), 'y, A005159_ser(N+14));
Vec(A238357_ser(11)) \\ Gheorghe Coserea, Jun 03 2017
A238358
Number of genus-8 rooted maps with n edges.
Original entry on oeis.org
11288163762500625, 2927974178219879250, 394372363395179602125, 36751560969705187643982, 2663973075006196131775590, 160098273686603663417293308, 8303278159618015743881266599, 381958851175370643701603049354, 15896435050196091382215375181044, 607566907750822335161584110201960
Offset: 16
Rooted maps with n edges of genus g for 0 <= g <= 10:
A000168,
A006300,
A006301,
A104742,
A215402,
A238355,
A238356,
A238357, this sequence,
A238359,
A238360.
-
T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 8];
Table[a[n], {n, 16, 30}] (* Jean-François Alcover, Jul 20 2018 *)
-
\\ see A238396
A238359
Number of genus-9 rooted maps with n edges.
Original entry on oeis.org
11665426077721040625, 3498878057690404966500, 540996834819906946713375, 57494374008560749302297480, 4724172886681078698955547790, 320061005837218582787265273000, 18618409220753939214291224549409, 956146512935178711199035220384800, 44232688287025023758415781081779828, 1871678026675570344184400604255444240
Offset: 18
Rooted maps with n edges of genus g for 0 <= g <= 10:
A000168,
A006300,
A006301,
A104742,
A215402,
A238355,
A238356,
A238357,
A238358, this sequence,
A238360.
-
T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 9];
Table[a[n], {n, 18, 30}] (* Jean-François Alcover, Jul 20 2018 *)
-
\\ see A238396
A238360
Number of genus-10 rooted maps with n edges.
Original entry on oeis.org
15230046989184655753125, 5199629454143883380553750, 909887917857275652461097750, 108861830345440643086946970900, 10021124647635764856828690342402, 757187906770815991999545249667404, 48918614114003431712044170834572688, 2779227352989564224315657269511192976, 141720718575991991799057452443438430580
Offset: 20
Rooted maps with n edges of genus g for 0 <= g <= 10:
A000168,
A006300,
A006301,
A104742,
A215402,
A238355,
A238356,
A238357,
A238358,
A238359, this sequence.
-
T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 10];
Table[a[n], {n, 20, 30}] (* Jean-François Alcover, Jul 20 2018 *)
-
\\ see A238396
A269920
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 0.
Original entry on oeis.org
1, 1, 1, 2, 5, 2, 5, 22, 22, 5, 14, 93, 164, 93, 14, 42, 386, 1030, 1030, 386, 42, 132, 1586, 5868, 8885, 5868, 1586, 132, 429, 6476, 31388, 65954, 65954, 31388, 6476, 429, 1430, 26333, 160648, 442610, 614404, 442610, 160648, 26333, 1430
Offset: 0
Triangle starts:
n\f [1] [2] [3] [4] [5] [6] [7] [8]
[0] 1;
[1] 1, 1;
[2] 2, 5, 2;
[3] 5, 22, 22, 5;
[4] 14, 93, 164, 93, 14;
[5] 42, 386, 1030, 1030, 386, 42;
[6] 132, 1586, 5868, 8885, 5868, 1586, 132;
[7] 429, 6476, 31388, 65954, 65954, 31388, 6476, 429;
[8] ...
-
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, 0], {n, 0, 8}, {f, 1, n+1}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)
-
N = 8; G = 0; 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))))
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
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] ...
-
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 *)
-
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))))
Showing 1-10 of 17 results.
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