A091401 -id:A091401 - cp's OEIS frontend

A273445 a(n) is the number of solutions of the equation n = A001617(k).

15, 12, 8, 11, 7, 14, 4, 13, 7, 12, 4, 15, 4, 9, 6, 10, 5, 16, 2, 20, 3, 14, 7, 11, 2, 13, 5, 11, 3, 14, 3, 9, 6, 13, 3, 17, 3, 14, 4, 10, 4, 20, 3, 15, 3, 12, 1, 15, 2, 20, 4, 11, 3, 13, 3, 16, 3, 12, 3, 15, 3, 12, 5, 9, 4, 15, 2, 14, 5, 17, 3, 13

Offset: 0

Gheorghe Coserea, May 22 2016

nonn

The zeros of the sequence are given by A054729. The first five zeros of the sequence have indexes 150, 180, 210, 286, 304.

			For n = 0 the a(0) = 15 solutions are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25 (A091401).
For n = 1 the a(1) = 12 solutions are:
11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49 (A091403).
For n = 2 the a(2) = 8 solutions are:
22, 23, 26, 28, 29, 31, 37, 50 (A091404).

Gheorghe Coserea, Table of n, a(n) for n = 0..100001
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.

Cf. A001617, A054729, A091401, A091403, A091404.

(* b = A001617 *) nmax = 71;
b[n_] := b[n] = If[n < 1, 0, 1 + Sum[ MoebiusMu[ d]^2 n/d / 12 - EulerPhi[ GCD[ d, n/d]] / 2, {d, Divisors[n]}] - Count[(#^2 - # + 1)/n& /@ Range[n], ?IntegerQ]/3 -Count[(#^2 + 1)/n& /@ Range[n], ?IntegerQ]/4];
Clear[f];
f[m_] := f[m] = Module[{}, A001617 = Array[b, m]; a[n_] := Count[A001617, n]; Table[a[n], {n, 0, nmax}]];
f[m = nmax]; f[m = m + nmax];
While[Print["m = ", m]; f[m] != f[m - nmax], m = m + nmax];
A273445 = f[m] (* Jean-François Alcover, Dec 16 2018, using Michael Somos' code for A001617 *)

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;
seq(n) = {
  my(a = vector(n+1,g,0), bnd = 12*n + 18*sqrtint(n) + 100, g);
  for (k = 1, bnd, g = A001617(k);
       if (g <= n, a[g+1]++));
  return(a);
};
seq(72)

a(n) = card {k, n = A001617(k)}.

A276019 n^2 * a(n) = (88n^2 - 132n + 54) * a(n-1) - 500(2n-3)^2 * a(n-2), with a(0)=1, a(1)=10.

Original entry on oeis.org

1, 10, 230, 6500, 199750, 6366060, 204990300, 6539387400, 202432551750, 5897526329500, 151804596385780, 2807347223915000, -15232296765302500, -5584390420089725000, -416025902106681525000, -24002385182809425846000, -1235898175219724085176250, -59486502796252242452122500, -2731496764897242177292037500, -120874274801920384164027025000, -5181210157044172846922944311500

Offset: 0

Gheorghe Coserea, Aug 23 2016

sign

			A(x) = 1 + 10*x + 230*x^2 + 6500*x^3 + ... is the g.f.

Gheorghe Coserea, Table of n, a(n) for n = 0..201
Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.

Cf. A091401, A276018.

seq(N) = {
  a = vector(N); a[1] = 10; a[2] = 230;
  for (n = 3, N, a[n] = ((88*n^2 - 132*n + 54)*a[n-1] - 500*(2*n-3)^2 * a[n-2])/n^2);
  concat(1, a);
};
seq(20)

n^2*a(n) = (88*n^2-132*n+54)*a(n-1) - 500*(2*n-3)^2*a(n-2), with a(0)=1, a(1)=10.

0 = 4*x*(x^2+22*x+125)*y'' + (8*x^2+132*x+500)*y' + (x+10)*y, where y(x) = A(x/-500).

A276020 n^2 * a(n) = 2(17n^2-21n+9) a(n-1) - 4(112n^2-280n+197) a(n-2) + 40(68n^2-256n+251) a(n-3) - 1600(2n-5)^2 * a(n-4), with a(0)=1, a(1)=10, a(2)=90, a(3)=780.

Original entry on oeis.org

1, 10, 90, 780, 6630, 55820, 469220, 3967000, 33951750, 295553500, 2622492940, 23701797800, 217528135900, 2018704327800, 18862262001800, 176834576018480, 1659586559786950, 15575074941839100, 146164364053448700, 1372547571923176200, 12910383388613518580, 121770360957324308200, 1152648798132152849400

Offset: 0

Gheorghe Coserea, Aug 23 2016

nonn

Gheorghe Coserea, Table of n, a(n) for n = 0..201
Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.

Cf. A091401, A276018.

a[0] = 1; a[1] = 10; a[2] = 90; a[3] = 780; a[n_] := a[n] = (40(68n^2 - 256n + 251)a[n-3] - 4(112n^2 - 280n + 197)a[n-2] + 2(17n^2 - 21n + 9)a[n-1] - 1600(2n - 5)^2 a[n-4])/n^2;
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Oct 19 2018 *)

seq(N) = {
  my(a = vector(N));
  a[1] = 10; a[2] = 90; a[3] = 780; a[4] = 6630;
  for (n = 5, N,
    my(t1 = 2*(17*n^2 - 21*n + 9)*a[n-1] - 4*(112*n^2 - 280*n + 197)*a[n-2],
       t2 = 40*(68*n^2 - 256*n + 251) * a[n-3] - 1600*(2*n-5)^2 *a[n-4]);
    a[n] = (t1 + t2)/n^2);
  concat(1,a);
};
seq(22)

n^2*a(n) = 2*(17*n^2-21*n+9)*a(n-1) - 4*(112*n^2-280*n+197)*a(n-2) + 40*(68*n^2-256*n+251)*a(n-3) - 1600*(2*n-5)^2 *a(n-4), with a(0)=1, a(1)=10, a(2)=90, a(3)=780.
0 = 4*x*(x+4)*(x+5)*(x^2+8*x+20)*y'' + 4*(4*x^4+55*x^3+280*x^2+600*x+400)*y' + (9*x^3+95*x^2+340*x+400)*y, where y(x) = A(x/-40).
a(n) ~ 2^n * 5^(n+5/4) / (Pi*n). - Vaclav Kotesovec, Aug 25 2016

A276021 n^2 * a(n) = 3(39n^2 - 52n + 20) a(n-1) - 441(3n-4)^2 * a(n-2), with a(0)=1, a(1)=21.

Original entry on oeis.org

1, 21, 693, 23940, 734643, 13697019, -494620749, -83079255420, -6814815765975, -444980496382695, -25071954462140859, -1226361084729855984, -49426887403935395172, -1287188243957889124740, 23935850133162849385308, 6798920856226697943604944, 650950202721260061404073891

Offset: 0

Gheorghe Coserea, Aug 23 2016

sign

			A(x) = 1 + 21*x + 693*x^2 + 23940*x^3 + ... is the g.f.

Gheorghe Coserea, Table of n, a(n) for n = 0..201
Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.

Cf. A091401, A276018.

I:=[21,693]; [1] cat [n le 2 select I[n] else (3*(39*n^2-52*n+20)*Self(n-1)-441*(3*n-4)^2*Self(n-2)) div n^2: n in [1..30]]; // Vincenzo Librandi, Aug 25 2016

a[0] = 1; a[1] = 21; a[n_] := a[n] = (3(39n^2 - 52n + 20) a[n-1] - 441(3n - 4)^2 a[n-2])/n^2;
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Oct 19 2018 *)

seq(N) = {
  my(a = vector(N)); a[1] = 21; a[2] = 693;
  for (n=3, N,
       a[n] = (3*(39*n^2 - 52*n + 20) * a[n-1] - 441*(3*n-4)^2 * a[n-2])/n^2);
  concat(1,a);
};
seq(17)

n^2 * a(n) = 3*(39*n^2-52*n+20) * a(n-1) - 441*(3*n-4)^2 * a(n-2), with a(0)=1, a(1)=21.

0 = 9*x*(x^2+13*x+49)*y'' + (21*x^2 + 195*x + 441)*y' + (4*x+21)*y, where y(x) = A(x/-441).

A276022 n^2 * a(n) = 3(5n^2 - 5n + 2) a(n-1) - 16(5n^2 - 10n + 6) a(n-2) + 36(5n^2 - 15n + 12) a(n-3) - 144(n-2)^2 a(n-4), with a(0)=1, a(1)=6, a(2)=30, a(3)=144.

Original entry on oeis.org

1, 6, 30, 144, 690, 3348, 16536, 83232, 426618, 2223180, 11756052, 62959680, 340881792, 1862954784, 10262937600, 56926831104, 317632207194, 1781352834300, 10034760283356, 56748881420640, 322033934657628, 1833043230774360, 10462349697348528, 59861990921495616

Offset: 0

Gheorghe Coserea, Aug 23 2016

nonn

			A(x) = 1 + 6*x + 30*x^2 + 144*x^3 + ... is the g.f.

Gheorghe Coserea, Table of n, a(n) for n = 0..301
Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.

Cf. A091401, A276018.

I:=[6,30,144,690]; [1] cat [n le 4 select I[n] else (3*(5*n^2-5*n+2)*Self(n-1)-16*(5*n^2-10*n+6)*Self(n-2)+36*(5*n^2-15*n+12)*Self(n-3)-144*(n-2)^2*Self(n-4)) div n^2: n in [1..30]]; // Vincenzo Librandi, Aug 25 2016

a[0] = 1; a[1] = 6; a[2] = 30; a[3] = 144; a[n_] := a[n] = (3(5n^2 - 5n + 2) a[n-1] - 16(5n^2 - 10n + 6)a[n-2] + 36(5n^2 - 15n + 12) a[n-3] - 144(n-2)^2 a[n-4])/n^2;
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Oct 19 2018 *)

seq(N) = {
  my(a = vector(N)); a[1] = 6; a[2] = 30; a[3] = 144; a[4] = 690;
  for (n=5, N,
       my(t1 = 3*(5*n^2 - 5*n + 2)*a[n-1] - 16*(5*n^2 - 10*n + 6)*a[n-2],
          t2 = 36*(5*n^2 - 15*n + 12)*a[n-3] - 144*(n-2)^2 * a[n-4]);
       a[n] = (t1+t2)/n^2);
  concat(1,a);
};
seq(25)

n^2 * a(n) = 3*(5*n^2 - 5*n + 2) * a(n-1) - 16*(5*n^2 - 10*n + 6) * a(n-2) + 36*(5*n^2 - 15*n + 12) * a(n-3) - 144*(n-2)^2 * a(n-4), with a(0)=1, a(1)=6, a(2)=30, a(3)=144.
0 = x*(x+2)*(x+3)*(x+4)*(x+6)*y'' + (5*x^4 + 60*x^3 + 240*x^2 + 360*x + 144)*y' + 4*(x^2+6*x+6)*(x+3)*y, where y(x) = A(x/-12).
a(n) ~ 2^n * 3^(n+3/2) / (Pi*n). - Vaclav Kotesovec, Aug 25 2016

A276177 n^2 * a(n) = 6(66n^2 - 94n + 41) a(n-1) - 36(2016n^2 - 5712n + 4387) a(n-2) + 50544(132n^2 - 560n + 609) a(n-3) - 7884864(6n-17)^2*a(n-4), with a(0)=1, a(1)=78, a(2)=4446, a(3)=20124.

Original entry on oeis.org

1, 78, 4446, 20124, -38185290, -6138851004, -560711991060, -21068540562888, 3057536757534246, 744702083933794740, 85203074089262120004, 5052846560269468159368, -180318018879496001303748, -86176724948835065345458008, -11276003918572185562671306600, -751248675388448553292016521104

Offset: 0

Gheorghe Coserea, Aug 23 2016

sign

			A(x) = 1 + 78*x + 4446*x^2 + 20124*x^3 + ... is the g.f.

Gheorghe Coserea, Table of n, a(n) for n = 0..301
Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.

Cf. A091401, A276018.

I:=[78,4446,20124,-38185290]; [1] cat [n le 4 select I[n] else (6*(66*n^2-94*n+41)*Self(n-1)-36*(2016*n^2-5712*n+4387)*Self(n-2)+50544*(132*n^2-560*n+ 609)*Self(n-3)-7884864*(6*n-17)^2*Self(n-4)) div n^2: n in [1..30]]; // Vincenzo Librandi, Aug 25 2016

a[0] = 1; a[1] = 78; a[2] = 4446; a[3] = 20124; a[n_] := a[n] = (6(66n^2 - 94n + 41) a[n-1] - 36(2016n^2 - 5712n + 4387)a[n-2] + 50544(132n^2 - 560n + 609)a[n-3] - 7884864(6n - 17)^2 a[n-4])/n^2;
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Oct 19 2018 *)

seq(N) = {
  my(a = vector(N));
  a[1] = 78; a[2] = 4446; a[3] = 20124; a[4] = -38185290;
  for (n = 5, N,
       my(t1 = 6*(66*n^2 - 94*n + 41) * a[n-1],
          t2 = -36*(2016*n^2 - 5712*n + 4387) * a[n-2],
          t3 = 50544*(132*n^2 - 560*n + 609) * a[n-3],
          t4 = -7884864*(6*n-17)^2 * a[n-4]);
       a[n] = (t1 + t2 + t3 + t4)/n^2);
  concat(1,a);
};
seq(17)

n^2 * a(n) = 6*(66*n^2 - 94*n + 41) * a(n-1) - 36*(2016*n^2 - 5712*n + 4387) * a(n-2) + 50544*(132*n^2 - 560*n + 609) * a(n-3) - 7884864*(6*n-17)^2*a(n-4), with a(0)=1, a(1)=78, a(2)=4446, a(3)=20124.

0 = 36*x*(x^2 + 5*x + 13)*(x^2 + 6*x + 13)*y'' + 12*(10*x^4 + 91*x^3 + 364*x^2 + 676*x + 507)*y' + (49*x^3 + 351*x^2 + 1027*x + 1014)*y, where y(x) = A(x/-468).

A276178 G.f.: 1/AGM(1, (1-4*x)^2).

Original entry on oeis.org

1, 4, 12, 32, 84, 240, 784, 2816, 10404, 38096, 137456, 493440, 1783376, 6532288, 24245568, 90814464, 341776164, 1289126160, 4870386736, 18439692928, 70004793936, 266551445952, 1017708956224, 3894679004160, 14932998810896, 57349426579264, 220574904103872, 849571289810432

Offset: 0

Gheorghe Coserea, Aug 23 2016

nonn

			A(x) = 1 + 4*x + 12*x^2 + 32*x^3 + ... is the g.f.

Gheorghe Coserea, Table of n, a(n) for n = 0..301
Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.

Cf. A091401, A276018.

a[n_] = DifferenceRoot[Function[{a, n}, {(-80 n^2 - 400n - 512) a[n+1] + (40n^2 + 240n + 368) a[n+2] + (-10n^2 - 70n - 124) a[n+3] + 64(n+2)^2 a[n] + (n+4)^2 a[n+4] == 0, a[0] == 1, a[1] == 4, a[2] == 12, a[3] == 32}]][n];
Table[a[n], {n, 0, 27}]
(* or: *)
Series[1/FunctionExpand[ArithmeticGeometricMean[1, (1-4x)^2], 1-4x > 0], {x, 0, 28}] // CoefficientList[#, x]& (* Jean-François Alcover, Dec 18 2018 *)

N=34; x='x + O('x^N); Vec(1/agm(1, (1-4*x)^2))

G.f.: 1/agm(1, (1-4*x)^2).
0 = x*(x+2)*(x+4)*(x^2 + 4*x + 8) * y'' + (5*x^4 + 40*x^3 + 120*x^2 + 160*x + 64) * y' + 4*(x+2)^3 * y, where y(x) = A(x/-8).
From Vaclav Kotesovec, Aug 25 2016: (Start)
Recurrence: n^2*a(n) = 2*(5*n^2 - 5*n + 2)*a(n-1) - 8*(5*n^2 - 10*n + 6)*a(n-2) + 16*(5*n^2 - 15*n + 12)*a(n-3) - 64*(n-2)^2*a(n-4).
a(n) ~ 2^(2*n+2)/(Pi*n).
(End)

A276179 n^2 * a(n) = 2(7n^2 - 7n + 3)a(n-1) - 12(7n^2 - 14n + 9)a(n-2) + 39(7n^2 - 21n + 18) a(n-3) - 72(7n^2 - 28n + 30)a(n-4) + 72(7n^2 - 35n + 45) a(n-5) - 216(n-3)^2 a(n-6), with a(0)=1, a(1)=6, a(2)=24, a(3)=78, a(4)=216, a(5)=504.

Original entry on oeis.org

1, 6, 24, 78, 216, 504, 906, 756, -2808, -17832, -57312, -104832, 81882, 1734156, 9360576, 35755956, 106475472, 232967664, 215497680, -1178534304, -8734303296, -36146763648, -108833048064, -220247838720, -46688571558, 2220777704700, 13473296923536, 53523581091900

Offset: 0

Gheorghe Coserea, Aug 24 2016

sign

			A(x) = 1 + 6*x + 24*x^2 + 78*x^3 + 216*x^4 + 504*x^5 + 906*x^6 + ... is the g.f.

Gheorghe Coserea, Table of n, a(n) for n = 0..301
Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.

Cf. A091401, A276018.

I:=[6,24,78,216,504,906]; [1] cat [n le 6 select I[n] else (2*(7*n^2-7*n+3)*Self(n-1)-12*(7*n^2-14*n+9)*Self(n-2)+39*(7*n^2-21*n+18)*Self(n-3)-72*(7*n^2-28*n+30)*Self(n-4)+72*(7*n^2-35*n+45)*Self(n-5)-216*(n-3)^2*Self(n-6)) div n^2: n in [1..30]]; // Vincenzo Librandi, Aug 25 2016

seq(N) = {
  my(a = vector(N));
  a[1] = 6; a[2] = 24; a[3] = 78; a[4] = 216; a[5] = 504; a[6] = 906;
  for (n = 7, N,
  my(t1 = 2*(7*n^2 - 7*n + 3)*a[n-1] - 12*(7*n^2 - 14*n + 9)*a[n-2],
     t2 = 39*(7*n^2 - 21*n + 18) * a[n-3] - 72*(7*n^2 - 28*n + 30)*a[n-4],
     t3 = 72*(7*n^2 - 35*n + 45) * a[n-5] - 216*(n-3)^2 * a[n-6]);
     a[n] = (t1+t2+t3)/n^2);
  concat(1,a);
};
seq(33)

n^2 * a(n) = 2*(7*n^2 - 7*n + 3)*a(n-1) - 12*(7*n^2 - 14*n + 9)*a(n-2) + 39*(7*n^2 - 21*n + 18) * a(n-3) - 72*(7*n^2 - 28*n + 30)*a(n-4) + 72*(7*n^2 - 35*n + 45) * a(n-5) - 216*(n-3)^2 * a(n-6), with a(0)=1, a(1)=6, a(2)=24, a(3)=78, a(4)=216, a(5)=504.

0 = x*(x+2)*(x+3)*(x^2+3*x+3)*(x^2+6*x+12)*y'' + (7*x^6 + 84*x^5 + 420*x^4 + 1092*x^3 + 1512*x^2 + 1008*x + 216)*y' + 9*(x+2)^2 * (x^3 + 6*x^2 + 12*x + 6)*y, where y(x) = A(x/-6).

A276180 n^2a(n) = 2(14n^2 - 16n + 7)a(n-1) - 20(24n^2 - 56n + 41)a(n-2) + 80(64n^2 - 224n + 221)a(n-3) - 1600(24n^2 - 112n + 139)a(n-4) + 6400(28n^2 - 164n + 245)a(n-5) - 128000(2n-7)^2a(n-6) for n>6, a(0)=1, a(1)=10, a(2)=30, a(3)=-300, a(4)=-3850, a(5)=-13940, a(6) = 56300.

Original entry on oeis.org

1, 10, 30, -300, -3850, -13940, 56300, 543400, -2332250, -29758500, 340835780, 7316239000, 40381709500, -199606565000, -4494519345000, -25429880846000, 18331676223750, 848074482677500, 714616060812500, -19019302889325000, 506727569992188500

Offset: 0

Gheorghe Coserea, Aug 24 2016

sign

			A(x) = 1 + 10*x + 30*x^2 - 300*x^3 - 3850*x^4 - 13940*x^5 + ... is the g.f.

Gheorghe Coserea, Table of n, a(n) for n = 0..301
Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.

Cf. A091401, A276018.

I:=[10,30,-300,-3850,-13940,56300]; [1] cat [n le 6 select I[n] else (2*(14*n^2-16*n+7)*Self(n-1)-20*(24*n^2- 56*n+41)*Self(n-2)+80*(64*n^2-224*n+221)*Self(n-3)-1600*(24*n^2-112*n+139)*Self(n-4)+6400*(28*n^2-164*n+245)*Self(n-5)-128000*(2*n-7)^2*Self(n-6))div n^2: n in [1..30]]; // Vincenzo Librandi, Aug 25 2016

a[n_] := a[n] = If[n <= 6, {1, 10, 30, -300, -3850, -13940, 56300}[[n+1]], (1/n^2)(6400(28n^2 - 164n + 245) a[n-5] - 1600(24n^2 - 112n + 139) a[n-4] + 80(64n^2 - 224n + 221) a[n-3] - 20(24n^2 - 56n + 41) a[n-2] + 2(14n^2 - 16n + 7) a[n-1] - 128000(2n - 7)^2 a[n-6])];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 16 2018 *)

seq(N) = {
  my(a = vector(N));
  a[1] = 10; a[2] = 30; a[3] = -300; a[4] = -3850; a[5] = -13940; a[6] = 56300;
  for (n=7, N,
  my(t1 = 2*(14*n^2 - 16*n + 7)*a[n-1] - 20*(24*n^2 - 56*n + 41)*a[n-2],
     t2 = 80*(64*n^2 - 224*n + 221)*a[n-3] - 1600*(24*n^2 - 112*n + 139)*a[n-4],
     t3 = 6400*(28*n^2 - 164*n + 245)*a[n-5] - 128000*(2*n-7)^2 * a[n-6]);
  a[n] = (t1+t2+t3)/n^2);
  concat(1,a);
};
seq(25)

n^2*a(n) = 2*(14*n^2 - 16*n + 7)*a(n-1) - 20*(24*n^2 - 56*n + 41)*a(n-2) + 80*(64*n^2 - 224*n + 221)*a(n-3) - 1600*(24*n^2 - 112*n + 139)*a(n-4) + 6400*(28*n^2 - 164*n + 245)*a(n-5) - 128000*(2*n-7)^2*a(n-6) for n>6, a(0)=1, a(1)=10, a(2)=30, a(3)=-300, a(4)=-3850, a(5)=-13940, a(6)=56300.

0 = 4*x*(x^2+2*x+5)*(x^4+5*x^3+15*x^2+25*x+25)*y'' + (24*x^6 + 144*x^5 + 520*x^4 + 1120*x^3 + 1600*x^2 + 1300*x + 500)*y' + 25*(x^5 + 5*x^4 + 15*x^3 + 25*x^2 + 25*x + 10)*y, where y(x) = A(x/-20).

A273510 a(n) is the largest level N such that genus of modular curve X_0(N) is n (or -1 if no such curve exists).

Original entry on oeis.org

25, 49, 50, 64, 81, 75, 121, 100, 169, 128, 127, 147, 157, 163, 181, 193, 199, 289, 229, 243, 239, 257, 361, 283, 293, 313, 343, 337, 349, 353, 373, 379, 397, 409, 421, 529, 439, 457, 463, 467, 487, 499, 509, 523, 541, 547, 557, 577, 625, 601, 613, 619, 631, 643, 661, 673, 677, 691, 841, 667, 733

Offset: 0

Gheorghe Coserea, May 23 2016

sign

a(10^7) = 120000007 is the largest value in the first 1+10^7 terms of the sequence.

The exception occurs first at a(150) = -1. - Georg Fischer, Feb 15 2019

			For n = 0 we have 0 = A001617(k) when k is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25 (A091401); the largest of this values is 25 therefore a(0) = 25.
For n = 1 we have 1 = A001617(k) when k is 11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49 (A091403); the largest of this values is 49 therefore a(1) = 49.
For n = 2 we have 2 = A001617(k) when k is 22, 23, 26, 28, 29, 31, 37, 50 (A091404); the largest of this values is 50 therefore a(2) = 50.
For n = 150 (= A054729(1)) we have 150 <> A001617(k) for all k therefore a(150) = -1.

Gheorghe Coserea, Table of n, a(n) for n = 0..200010
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.

Cf. A001617, A054728, A054729, A091401, A091403, A091404, A273445.

a1617[n_] := If[n < 1, 0, 1 + Sum[MoebiusMu[d]^2 n/d/12 - EulerPhi[GCD[d, n/d]]/2, {d, Divisors[n]}] - Count[(#^2 - # + 1)/n& /@ Range[n], ?IntegerQ]/3 - Count[(#^2 + 1)/n& /@ Range[n], ?IntegerQ]/4];
seq[n_] := Module[{a, bnd}, a = Table[-1, {n+1}]; bnd = 12n + 18 Floor[Sqrt[n] ] + 100; For[k = 1, k <= bnd, k++, g = a1617[k]; If[g <= n, a[[g+1]] = k]]; a];
seq[60] (* Jean-François Alcover, Nov 20 2018, after Gheorghe Coserea and Michael Somos in A001617 *)

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;
seq(n) = {
  my(a = vector(n+1, g, -1), bnd = 12*n + 18*sqrtint(n) + 100, g);
  for (k = 1, bnd, g = A001617(k); if (g <= n, a[g+1] = k));
  return(a);
};
seq(60)

Let S(n) = {k, n = A001617(k)}; if the level set S(n) is not empty then a(n) = max S(n) and A054728(n) = min S(n) and A273445(n) = card S(n), otherwise a(n) = A054728(n) = -1 and A273445(n) = 0.

cp's OEIS Frontend

A273445 a(n) is the number of solutions of the equation n = A001617(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A276019 n^2 * a(n) = (88*n^2 - 132*n + 54) * a(n-1) - 500*(2*n-3)^2 * a(n-2), with a(0)=1, a(1)=10.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A276020 n^2 * a(n) = 2*(17*n^2-21*n+9) * a(n-1) - 4*(112*n^2-280*n+197) * a(n-2) + 40*(68*n^2-256*n+251) * a(n-3) - 1600*(2*n-5)^2 * a(n-4), with a(0)=1, a(1)=10, a(2)=90, a(3)=780.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A276021 n^2 * a(n) = 3*(39*n^2 - 52*n + 20) * a(n-1) - 441*(3*n-4)^2 * a(n-2), with a(0)=1, a(1)=21.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A276022 n^2 * a(n) = 3*(5*n^2 - 5*n + 2) * a(n-1) - 16*(5*n^2 - 10*n + 6) * a(n-2) + 36*(5*n^2 - 15*n + 12) * a(n-3) - 144*(n-2)^2 * a(n-4), with a(0)=1, a(1)=6, a(2)=30, a(3)=144.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A276177 n^2 * a(n) = 6*(66*n^2 - 94*n + 41) * a(n-1) - 36*(2016*n^2 - 5712*n + 4387) * a(n-2) + 50544*(132*n^2 - 560*n + 609) * a(n-3) - 7884864*(6*n-17)^2*a(n-4), with a(0)=1, a(1)=78, a(2)=4446, a(3)=20124.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A276178 G.f.: 1/AGM(1, (1-4*x)^2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A276019 n^2 * a(n) = (88n^2 - 132n + 54) * a(n-1) - 500(2n-3)^2 * a(n-2), with a(0)=1, a(1)=10.

A276020 n^2 * a(n) = 2(17n^2-21n+9) a(n-1) - 4(112n^2-280n+197) a(n-2) + 40(68n^2-256n+251) a(n-3) - 1600(2n-5)^2 * a(n-4), with a(0)=1, a(1)=10, a(2)=90, a(3)=780.

A276021 n^2 * a(n) = 3(39n^2 - 52n + 20) a(n-1) - 441(3n-4)^2 * a(n-2), with a(0)=1, a(1)=21.

A276022 n^2 * a(n) = 3(5n^2 - 5n + 2) a(n-1) - 16(5n^2 - 10n + 6) a(n-2) + 36(5n^2 - 15n + 12) a(n-3) - 144(n-2)^2 a(n-4), with a(0)=1, a(1)=6, a(2)=30, a(3)=144.

A276177 n^2 * a(n) = 6(66n^2 - 94n + 41) a(n-1) - 36(2016n^2 - 5712n + 4387) a(n-2) + 50544(132n^2 - 560n + 609) a(n-3) - 7884864(6n-17)^2*a(n-4), with a(0)=1, a(1)=78, a(2)=4446, a(3)=20124.

A276179 n^2 * a(n) = 2(7n^2 - 7n + 3)a(n-1) - 12(7n^2 - 14n + 9)a(n-2) + 39(7n^2 - 21n + 18) a(n-3) - 72(7n^2 - 28n + 30)a(n-4) + 72(7n^2 - 35n + 45) a(n-5) - 216(n-3)^2 a(n-6), with a(0)=1, a(1)=6, a(2)=24, a(3)=78, a(4)=216, a(5)=504.