A276018 -id:A276018 - cp's OEIS frontend

A091401 Numbers n such that genus of group Gamma_0(n) is zero.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25

Offset: 1

N. J. A. Sloane, Mar 02 2004

Equivalently, numbers n such that genus of modular curve X_0(n) is zero.

G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see Prop. 1.40 and 1.43.

Miranda C. N. Cheng, John F. R. Duncan and Jeffrey A Harvey, Umbral moonshine and the Niemeier lattices, Research in the Mathematical Sciences, 2014, 1:3; See Eq. (22). - _N. J. A. Sloane_, Jun 19 2014
K. Harada, "Moonshine" of Finite Groups, European Math. Soc., 2010, p. 15.
Yang-Hui He, John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.
Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.
K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series, CBMS Regional Conference Series in Mathematics, vol. 102, American Mathematical Society, Providence, RI, 2004. See p. 110.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 103.

Cf. A001617, A001615, A000089, A000086, A001616, A091403.
The table below is a consequence of Theorem 7.3 in Maier's paper.
N        EntryID        K       alpha
1
      A127776        4096    1
      A276018        729     1
      A002894        256     1
      A276019        125     4
      A093388        72      1
      A276021        49      9
      A081085        32      1
      A006077        27      1
     A276020        20      2
     A276022        12      1
     A276177        13      36
     A276178        8       1
     A276179        6       1
     A276180        5       4

Flatten@ Position[#, 0] &@ Table[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], {n, 120}] (* Michael De Vlieger, Dec 05 2016, after Michael Somos at A001617 *)

Numbers n such that A001617(n) = 0.

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).

cp's OEIS Frontend

A091401 Numbers n such that genus of group Gamma_0(n) is zero.

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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.

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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.

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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.

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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.

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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.

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

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A276179 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.

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.