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%I M4058 N1683 #97 Mar 22 2022 05:53:41
%S 1,-6,6,4,6,12,28,72,198,572,1716,5304,16796,54264,178296,594320,
%T 2005830,6843420,23571780,81880920,286583220,1009864680,3580429320,
%U 12765008880,45741281820,164668614552,595340375688,2160865067312,7871722745208,28772503827312
%N Expansion of (1-4*x)^(3/2) in powers of x.
%C Terms that are not divisible by 12 have indices in A019469. - _Ralf Stephan_, Aug 26 2004
%C From _Ralf Steiner_, Apr 06 2017: (Start)
%C By analytic continuation to the entire complex plane there exist regularized values for divergent sums such as:
%C Sum_{k>=0} a(k)^2/8^k = 2F1(-3/2,-3/2,1,2).
%C Sum_{k>=0} a(k) / 2^k = -i. (End)
%D A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 55.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 164.
%H Vincenzo Librandi, <a href="/A002421/b002421.txt">Table of n, a(n) for n = 0..1000</a>
%H N. J. A. Sloane, <a href="/A000984/a000984.pdf">Notes on A984 and A2420-A2424</a>
%F a(n) = Sum_{m=0..n} binomial(n, m)*K_m(4), where K_m(x) = K_m(n, 2, x) is a Krawtchouk polynomial. - Alexander Barg (abarg(AT)research.bell-labs.com)
%F a(n) ~ (3/4)*Pi^(-1/2)*n^(-5/2)*2^(2*n)*(1 + 15/8*n^-1 + ...). - Joe Keane (jgk(AT)jgk.org), Nov 22 2001
%F From _Ralf Stephan_, Mar 11 2004: (Start)
%F a(n) = 12*(2*n-4)! /(n!*(n-2)!), n > 1.
%F a(n) = 12*Cat(n-2)/n = 2(Cat(n-1) - 4*Cat(n-2)), in terms of Catalan numbers (A000108).
%F Terms that are not divisible by 12 have indices in A019469. (End)
%F Let rho(x)=(1/Pi)*(x*(4-x))^(3/2), then for n >= 4, a(n) = Integral_{x=0..4} (x^(n-4) *rho(x)) dx. - _Groux Roland_, Mar 16 2011
%F G.f.: (1-4*x)^(3/2) = 1 - 6*x + 12*x^2/(G(0) + 2*x); G(k) = (4*x+1)*k-2*x+2-2*x*(k+2)*(2*k+1)/G(k+1); for -1/4 <= x < 1/4, otherwise G(0) = 2*x; (continued fraction). - _Sergei N. Gladkovskii_, Dec 05 2011
%F G.f.: 1/G(0) where G(k) = 1 + 4*x*(2*k+1)/(1 - 1/(1 + (2*k+2)/G(k+1))); (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Nov 18 2012
%F G.f.: G(0)/2, where G(k) = 2 + 2*x*(2*k-3)*G(k+1)/(k+1). - _Sergei N. Gladkovskii_, Jun 06 2013 [Edited by _Michael Somos_, Dec 04 2013]
%F 0 = a(n+2) * (a(n+1) - 14*a(n)) + a(n+1) * (6*a(n+1) + 16*a(n)) for all n in Z. - _Michael Somos_, Dec 04 2013
%F A232546(n) = 3^n * a(n). - _Michael Somos_, Dec 04 2013
%F G.f.: hypergeometric1F0(-3/2;;4*x). - _R. J. Mathar_, Aug 09 2015
%F a(n) = 3*4^(n-1)*Gamma(-3/2+n)/(sqrt(Pi)*Gamma(1+n)). - _Peter Luschny_, Dec 14 2015
%F From _Ralf Steiner_, Apr 06 2017: (Start)
%F Sum_{k>=0} a(k)/4^k = 0.
%F Sum_{k>=0} a(k)^2/16^k = 32/(3*Pi).
%F Sum_{k>=0} a(k)^2*(k/8)/16^k = 1/Pi.
%F Sum_{k>=0} a(k)^2*(-k/24+1/8)/16^k = 1/Pi.
%F Sum_{k>=0} a(k-1)^2*(k-1/4)/16^k = 1/Pi.
%F Sum_{k>=0} a(k-1)^2*(2k-2)/16^k = 1/Pi.(End)
%F D-finite with recurrence: n*a(n) +2*(-2*n+5)*a(n-1)=0. - _R. J. Mathar_, Feb 20 2020
%F From _Amiram Eldar_, Mar 22 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 4/3 + 10*Pi/(81*sqrt(3)).
%F Sum_{n>=0} (-1)^n/a(n) = 92/75 - 4*sqrt(5)*log(phi)/125, where phi is the golden ratio (A001622). (End)
%e G.f. = 1 - 6*x + 6*x^2 + 4*x^3 + 6*x^4 + 12*x^5 + 28*x^6 + 72*x^7 + 198*x^8 + 572*x^9 + ...
%p A002421 := n -> 3*4^(n-1)*GAMMA(-3/2+n)/(sqrt(Pi)*GAMMA(1+n)):
%p seq(A002421(n), n=0..29); # _Peter Luschny_, Dec 14 2015
%t CoefficientList[Series[(1-4x)^(3/2),{x,0,40}],x] (* _Vincenzo Librandi_, Jun 11 2012
%t a[n_]:= Binomial[ 3/2, n] (-4)^n; (* _Michael Somos_, Dec 04 2013 *)
%t a[n_]:= SeriesCoefficient[(1-4x)^(3/2), {x, 0, n}]; (* _Michael Somos_, Dec 04 2013 *)
%o (Magma) [1,-6] cat [12*Catalan(n-2)/n: n in [2..30]]; // _Vincenzo Librandi_, Jun 11 2012
%o (PARI) {a(n) = binomial( 3/2, n) * (-4)^n}; /* _Michael Somos_, Dec 04 2013 */
%o (PARI) {a(n) = if( n<0, 0, polcoeff( (1 - 4*x + x * O(x^n))^(3/2), n))}; /* _Michael Somos_, Dec 04 2013 */
%o (Sage) ((1-4*x)^(3/2)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 03 2019
%o (GAP) Concatenation([1], List([1..40], n-> 12*Factorial(2*n-4) /( Factorial(n)*Factorial(n-2)) )) # _G. C. Greubel_, Jul 03 2019
%Y Cf. A007054, A004001, A002420, A002422, A002423, A002424.
%Y Cf. A000257, A001622, A071721, A071724, A085687.
%K sign,easy
%O 0,2
%A _N. J. A. Sloane_
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