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A002421 Expansion of (1-4*x)^(3/2) in powers of x.
(Formerly M4058 N1683)
13
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, 164668614552, 595340375688, 2160865067312, 7871722745208, 28772503827312 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Terms that are not divisible by 12 have indices in A019469. - Ralf Stephan, Aug 26 2004

From Ralf Steiner, Apr 06 2017: (Start)

By analytic continuation to the entire complex plane there exist regularized values for divergent sums such as:

Sum_{k>=0} a(k)^2/8^k = 2F1(-3/2,-3/2,1,2).

Sum_{k>=0} a(k) / 2^k = -i. (End)

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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

N. J. A. Sloane, Notes on A984 and A2420-A2424

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. - Alexander Barg (abarg(AT)research.bell-labs.com)

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

From Ralf Stephan, Mar 11 2004: (Start)

a(n) = 12*(2*n-4)! /(n!*(n-2)!), n > 1.

a(n) = 12*Cat(n-2)/n = 2(Cat(n-1) - 4*Cat(n-2)), in terms of Catalan numbers (A000108).

Terms that are not divisible by 12 have indices in A019469. (End)

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)). - 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, otherwise G(0)= 2*x; (continued fraction). - Sergei N. Gladkovskii, Dec 05 2011

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

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]

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

A232546(n) = 3^n * a(n). - Michael Somos, Dec 04 2013

G.f.: hypergeometric1F0(-3/2;;4*x). - R. J. Mathar, Aug 09 2015

a(n) = 3*4^(n-1)*Gamma(-3/2+n)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015

From Ralf Steiner, Apr 06 2017: (Start)

Sum_{k>=0} a(k)/4^k = 0.

Sum_{k>=0} a(k)^2/16^k = 32/(3*Pi).

Sum_{k>=0} a(k)^2*(k/8)/16^k = 1/Pi.

Sum_{k>=0} a(k)^2*(-k/24+1/8)/16^k = 1/Pi.

Sum_{k>=0} a(k-1)^2*(k-1/4)/16^k = 1/Pi.

Sum_{k>=0} a(k-1)^2*(2k-2)/16^k = 1/Pi.(End)

D-finite with recurrence: n*a(n) +2*(-2*n+5)*a(n-1)=0. - R. J. Mathar, Feb 20 2020

EXAMPLE

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 + ...

MAPLE

A002421 := n -> 3*4^(n-1)*GAMMA(-3/2+n)/(sqrt(Pi)*GAMMA(1+n)):

seq(A002421(n), n=0..29); # Peter Luschny, Dec 14 2015

MATHEMATICA

CoefficientList[Series[(1-4x)^(3/2), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 11 2012

a[n_]:= Binomial[ 3/2, n] (-4)^n; (* Michael Somos, Dec 04 2013 *)

a[n_]:= SeriesCoefficient[(1-4x)^(3/2), {x, 0, n}]; (* Michael Somos, Dec 04 2013 *)

PROG

(MAGMA) [1, -6] cat [12*Catalan(n-2)/n: n in [2..30]]; // Vincenzo Librandi, Jun 11 2012

(PARI) {a(n) = binomial( 3/2, n) * (-4)^n}; /* Michael Somos, Dec 04 2013 */

(PARI) {a(n) = if( n<0, 0, polcoeff( (1 - 4*x + x * O(x^n))^(3/2), n))}; /* Michael Somos, Dec 04 2013 */

(Sage) ((1-4*x)^(3/2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 03 2019

(GAP) Concatenation([1], List([1..40], n-> 12*Factorial(2*n-4) /( Factorial(n)*Factorial(n-2)) )) # G. C. Greubel, Jul 03 2019

CROSSREFS

Cf. A007054, A004001, A002420, A002422, A002423, A002424.

Cf. A000257, A071721, A071724, A085687.

Sequence in context: A255438 A200491 A245297 * A209938 A165953 A045885

Adjacent sequences:  A002418 A002419 A002420 * A002422 A002423 A002424

KEYWORD

sign,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 22 12:53 EDT 2021. Contains 343177 sequences. (Running on oeis4.)