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A002421
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Expansion of (1-4x)^(3/2).
(Formerly M4058 N1683)
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7
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1, -6, 6, 4, 6, 12, 28, 72, 198, 572, 1716, 5304, 16796, 54264, 178296, 594320, 2005830, 6843420, 23571780, 81880920, 286583220, 1009864680, 3580429320, 12765008880, 45741281820
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Terms that are not divisible by 12 have indices in A019469. - R. Stephan, Aug 26 2004
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REFERENCES
| 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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 164.
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FORMULA
| 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 - abarg(AT)research.bell-labs.com (Alexander Barg).
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
For n>1, a(n) = 12 * (2n-4)! / [n!(n-2)! ] = 2(Cat(n-1)-4*Cat(n-2)) = 12*Cat(n-2)/n. Proof: G.f. is (1-4x) times the g.f. of A002420. - R. Stephan, Aug 26 2004
12 * (2n-4)! / [n(n-1)!(n-2)! ], n>1. In terms of Catalan numbers (A000108), a(n) = 12*Cat(n-2)/n. Terms that are not divisible by 12 have indices in A019469. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 11 2004
Let rho(x)=(1/Pi)*(x*(4-x))^(3/2), then for n>=4 a(n)=int(x^(n-4)*rho(x),x=0..4) - Groux Roland, Mar 16 2011
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 ,else G(0)= 2*x ; (continued fraction). - Sergei N. Gladkovskii, Dec 05 2011
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CROSSREFS
| Cf. A007054, A004001, A002420, A002422-A002424.
Cf. A000257, A071721, A071724, A085687.
Sequence in context: A099405 A090966 A200491 * A165953 A045885 A019118
Adjacent sequences: A002418 A002419 A002420 * A002422 A002423 A002424
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KEYWORD
| sign,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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