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] ############################################################################################## 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] ################################### e.o.f.####################################################