A381779
G.f. A(x) satisfies A(x) = (1 + x*A(x)) * C(x*A(x)^3), where C(x) is the g.f. of A000108.
Original entry on oeis.org
1, 2, 11, 95, 977, 11028, 132029, 1646428, 21155077, 278127359, 3723466202, 50586670945, 695676081162, 9665426437561, 135464096419620, 1912922793362142, 27190770354633287, 388734441118885467, 5586079818959767743, 80638973170989453862, 1168864771263296930809
Offset: 0
-
a(n) = sum(k=0, n, binomial(n+4*k+1, k)*binomial(n+2*k+1, n-k)/(n+4*k+1));
A381881
Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * C(x)) ), where C(x) is the g.f. of A000108.
Original entry on oeis.org
1, 3, 14, 82, 547, 3958, 30249, 240362, 1966235, 16449495, 140093989, 1210575512, 10587490383, 93540456103, 833619150838, 7484887130882, 67645312129491, 614872423359187, 5617522739173495, 51556112664387720, 475105557839611760, 4394434006611790855
Offset: 0
A054730
Odd n such that genus of modular curve X_0(N) is never equal to n.
Original entry on oeis.org
49267, 74135, 94091, 96463, 102727, 107643, 118639, 138483, 145125, 181703, 182675, 208523, 221943, 237387, 240735, 245263, 255783, 267765, 269627, 272583, 277943, 280647, 283887, 286815, 309663, 313447, 322435, 326355, 336675, 347823, 352719
Offset: 1
Janos A. Csirik, Apr 21 2000
- J. A. Csirik, The genus of X_0(N) is not 150, preprint, 2000.
-
A000089(n) = {
if (n%4 == 0 || n%4 == 3, return(0));
if (n%2 == 0, n \= 2);
my(f = factor(n), fsz = matsize(f)[1]);
prod(k = 1, fsz, if (f[k, 1] % 4 == 3, 0, 2));
};
A000086(n) = {
if (n%9 == 0 || n%3 == 2, return(0));
if (n%3 == 0, n \= 3);
my(f = factor(n), fsz = matsize(f)[1]);
prod(k = 1, fsz, if (f[k, 1] % 3 == 2, 0, 2));
};
A001615(n) = {
my(f = factor(n), fsz = matsize(f)[1],
g = prod(k=1, fsz, (f[k, 1]+1)),
h = prod(k=1, fsz, f[k, 1]));
return((n*g)\h);
};
A001616(n) = {
my(f = factor(n), fsz = matsize(f)[1]);
prod(k = 1, fsz, f[k, 1]^(f[k, 2]\2) + f[k, 1]^((f[k, 2]-1)\2));
};
A001617(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;
scan(n) = {
my(inv = vector(n+1, g, -1), bnd = 12*n + 18*sqrtint(n) + 100, g);
for (k = 1, bnd, g = A001617(k);
if (g <= n && inv[g+1] == -1, inv[g+1] = k));
select(x->(x%2==1), apply(x->(x-1), Vec(select(x->x==-1, inv, 1))));
};
scan(400*1000)
A200757
Noncrossing forests in the regular (n+1)-polygon obtained by a grafting procedure.
Original entry on oeis.org
1, 3, 13, 68, 395, 2450, 15892, 106489, 731379, 5121392, 36425796, 262425982, 1911063188, 14044679173, 104030937139, 775856119012, 5821085551579, 43906627941144, 332742274685104, 2532358764929916, 19346427410500788, 148312939031577504, 1140578980645677208
Offset: 1
-
f:= proc(n) option remember; local F;
if n=0 then 0 else F:= f(n-1);
convert(series(x+(x+F)^2/(1-x-F)-F^2/(1-F), x, n+1), polynom) fi
end:
a:= n-> coeff(f(n), x, n):
seq(a(n), n=1..30); # Alois P. Heinz, Nov 22 2011
-
a[n_] := Sum[ Binomial[3*n - i - 2, 2*n - 1]* Sum[Binomial[j, -2*n + 2*j + i]*(-1)^(n - j)*Binomial[n, j], {j, 0, n}], {i, 0, n - 1}]/n ; Table[a[n], {n, 1, 23}] (* Jean-François Alcover, Feb 22 2013, after Vladimir Kruchinin *)
-
a(n):=sum(binomial(3*n-i-2,2*n-1)*sum(binomial(j,-2*n+2*j+i)*(-1)^(n-j)*binomial(n,j),j,0,n),i,0,n-1)/n; /* Vladimir Kruchinin, Nov 25 2011 */
-
def suite_ncf(N):
ano = PowerSeriesRing(QQ,'x')
x = ano.gen()
F = ano.zero().O(1)
for k in range(N):
F = x+((x+F)**2/(1-x-F)-F**2/(1-F))
return F.O(N+1)
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