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
Triangle starts:
n\f [1] [2] [3] [4]
[8] 225225;
[9] 12317877, 12317877;
[10] 351683046, 792534015, 351683046;
[11] 7034538511, 26225260226, 26225260226, 7034538511;
[12] ...
Cf.
A035309,
A269921,
A269922,
A269923,
A269925,
A270406,
A270407,
A270408,
A270409,
A270410,
A270412.
-
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 *)
-
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
Triangle starts:
n\f [1] [2] [3] [4]
[10] 59520825;
[11] 4304016990, 4304016990;
[12] 15895975422, 354949166565, 158959754226;
[13] 4034735959800, 14805457339920, 14805457339920, 4034735959800;
[14] ...
Cf.
A035309,
A269921,
A269922,
A269923,
A269924,
A270406,
A270407,
A270408,
A270409,
A270410,
A270411,
A270412.
-
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 *)
-
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))))
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
-
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))
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
-
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 *)
-
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))
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