Rationals N2(n):=2*A124399(n)/A069955(n)

 Integral over square of monic Legendre polynomials 

 p(n,x):=(((2^n)*(n!)^2)/(2*n)!)*P(n,x) with the Legendre polynomial P(n,x).

 over the interval [-1,+1].

 Minimal norm square for real monic polynomials of degree n (cf. Courant-Hilbert reference).


  N2(n), n=0..30:

  [2, 2/3, 8/45, 8/175, 128/11025, 128/43659, 512/693693, 512/2760615, 32768/703956825, 

   32768/2807136475, 131072/44801898141, 131072/178837328943, 2097152/11425718238025, 

   2097152/45635265151875, 8388608/729232910488125, 8388608/2913690606794775, 

   2147483648/2980705490751054825, 2147483648/11912508103174630875, 

   8589934592/190453061649520333125, 8589934592/761284675790187924375, 

   137438953472/48691767863540419643025, 137438953472/194656659282135509820075, 

   549755813888/3112897815792828194230125, 549755813888/12445706768245428413604375, 

   35184372088832/3184718076363246848503430625, 35184372088832/12733776756530806199056117011, 

   140737488355328/203665080313034137018039551957, 140737488355328/814380945284628956663354312695,
   2251799813685248/52103760478924730186522770822425, 

   2251799813685248/208353087384808403825536359424275, 

   9007199254740992/3332723384435224201636023811413181] 

 
  Rationals  N2(n)/2, n=0..30:

  [1, 1/3, 4/45, 4/175, 64/11025, 64/43659, 256/693693, 256/2760615, 

  16384/703956825, 16384/2807136475, 65536/44801898141, 65536/178837328943, 

  1048576/11425718238025, 1048576/45635265151875, 4194304/729232910488125, 

  4194304/2913690606794775, 1073741824/2980705490751054825,

  1073741824/11912508103174630875, 4294967296/190453061649520333125, 

  4294967296/761284675790187924375, 68719476736/48691767863540419643025,

  68719476736/194656659282135509820075, 274877906944/3112897815792828194230125,
 
  274877906944/12445706768245428413604375, 17592186044416/3184718076363246848503430625,

  17592186044416/12733776756530806199056117011, 70368744177664/203665080313034137018039551957,

  70368744177664/814380945284628956663354312695,

  1125899906842624/52103760478924730186522770822425, 

  1125899906842624/208353087384808403825536359424275, 

  4503599627370496/3332723384435224201636023811413181] 

 
  Numerators of N2(n)/2 give A124399(n). For n=0..30:

  [1, 1, 4, 4, 64, 64, 256, 256, 16384, 16384, 65536, 65536, 1048576, 1048576, 4194304, 4194304, 

   1073741824, 1073741824, 4294967296, 4294967296, 68719476736, 68719476736, 274877906944, 

   274877906944, 17592186044416, 17592186044416, 70368744177664, 70368744177664, 

   1125899906842624, 1125899906842624, 4503599627370496]


  Denominators of N2(n)/2  give A069955(n). For n=0..30: 
  
  [1, 3, 45, 175, 11025, 43659, 693693, 2760615, 703956825, 2807136475, 44801898141, 

  178837328943, 11425718238025, 45635265151875, 729232910488125, 2913690606794775, 

  2980705490751054825, 11912508103174630875, 190453061649520333125, 761284675790187924375, 

  48691767863540419643025]


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  The values of N2(n), the minimal integral norm squares for real monic polynomials over 

  the interval [-1,+1] (attained exactly for the monic Legendre polynomials) are, 

  for n=0..20 (10 digits  Maple10; e-n means *10^(-n))

  [2., .6666666667, .1777777778, 0.4571428571e-1, 0.1160997732e-1, 0.2931812456e-2, 

   0.7380786602e-3, 0.1854659197e-3, 0.4654830927e-4, 0.1167310542e-4, 0.2925590331e-5, 

   0.7329118634e-6, 0.1835466232e-6, 0.4595463603e-7, 0.1150333162e-7, 0.2879031830e-8, 

   0.7204615332e-9, 0.1802713274e-9, 0.4510263326e-10, 0.1128347235e-10, 0.2822632233e-11] 

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