|
%I #23 Mar 12 2024 02:42:04
%S 1,96,14336,2523136,484442112,98180268032,20645907791872,
%T 4459516083044352,983075545417777152,220208922173582082048,
%U 49967406340478261526528,11459191854083014643417088,2651480699775516003646046208,618173786004806016850049630208
%N Expansion of (1 - (1-256*x)^(1/4)) / (64*x).
%H G. C. Greubel, <a href="/A254286/b254286.txt">Table of n, a(n) for n = 0..250</a>
%F G.f.: (1 - (1-256*x)^(1/4)) / (64*x).
%F a(n) ~ 256^n / (Gamma(3/4) * n^(5/4)).
%F Recurrence: (n+1)*a(n) = 64*(4*n-1)*a(n-1).
%F a(n) = 256^n * Gamma(n+3/4) / (Gamma(3/4) * Gamma(n+2)).
%F E.g.f.: hypergeom([3/4], [2], 256*x). - _Vaclav Kotesovec_, Jan 28 2015
%F From _Peter Bala_, Sep 01 2017: (Start)
%F a(n) = (-1)^n*binomial(1/4, n+1)*4^(4*n+1). Cf. A000108(n) = (-1)^n*binomial(1/2, n+1)*2^(2*n+1).
%F a(n) = 16^n*A025749(n+1); a(n) = 32^n*A048779(n+1).
%F (End)
%t CoefficientList[Series[(1-(1-256*x)^(1/4)) / (64*x),{x,0,20}],x]
%t CoefficientList[Series[Hypergeometric1F1[3/4,2,256*x],{x,0,20}],x] * Range[0,20]! (* _Vaclav Kotesovec_, Jan 28 2015 *)
%o (Magma) [Round(2^(8*n)*Gamma(n+3/4)/(Gamma(3/4)*Gamma(n+2))): n in [0..30]]; // _G. C. Greubel_, Aug 10 2022
%o (SageMath) [2^(8*n)*rising_factorial(3/4,n)/factorial(n+1) for n in (0..30)] # _G. C. Greubel_, Aug 10 2022
%Y Cf. A000108, A008545, A025749, A048779, A254282, A254287.
%K nonn,easy
%O 0,2
%A _Vaclav Kotesovec_, Jan 27 2015
| 