A232546 Expansion of (1 - 12*x)^(3/2) in powers of x.

1, -18, 54, 108, 486, 2916, 20412, 157464, 1299078, 11258676, 101328084, 939587688, 8926083036, 86514343272, 852784240824, 8527842408240, 86344404383430, 883760374277460, 9132190534200420, 95167038198509640, 999253901084351220, 10563541240034570040

Offset: 0

Author

Michael Somos, Nov 25 2013

Keywords

• sign

Comments

From Ralf Steiner, Apr 04 2017: (Start)

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

Sum_{k>=0} a(k)^2/16^k = 2F1(-3/2,-3/2,1,9).

Sum_{k>=0} a(k) / 6^k = -i. (End)

Examples

			G.f. = 1 - 18*x + 54*x^2 + 108*x^3 + 486*x^4 + 2916*x^5 + 20412*x^6 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..930
• R. Steiner, Sums of OEIS-A232546, ResearchGate, 2017. - _Ralf Steiner_, Apr 04 2017

Crossrefs

Cf. A000168, A002421, A115903.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (1 - 12 x)^(3/2), {x, 0, n}];
Table[9/Sqrt[Pi] 12^n Gamma[-1/2 + n]/Gamma[2 + n], {n, -1, 20}] (* Ralf Steiner, Apr 01 2017 *)
Flatten[{1, -18, Table[4*3^(n+1)*(2*n-4)!/((n-2)!*n!), {n, 2, 25}]}] (* Vaclav Kotesovec, Apr 02 2017 *)

PARI

{a(n) = if( n<0, 0, polcoeff( (1 - 12 * x + x * O(x^n))^(3/2), n))};

Formula

0 = a(n+2)*(a(n+1) - 42*a(n)) + 18*a(n+1)*(a(n+1) + 8*a(n)) for all n in Z.

a(n+2) = 54 * A000168(n). a(n) = 3^n * A002421(n). Convolution inverse of A115903.

a(n) = 6*(2*n-5)*a(n-1)/n. - R. J. Mathar, Nov 23 2014

G.f.: 1F0(-3/2;;12x). - R. J. Mathar, Aug 09 2015

For n>=2, a(n) = 4*3^(n+1)*(2*n-4)! / ((n-2)!*n!). - Vaclav Kotesovec, Apr 02 2017

Sum_{k>=0} a(k) / 12^k = 0. - Ralf Steiner, Apr 04 2017