OFFSET
0,2
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)
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
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) = 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
EXAMPLE
G.f. = 1 - 18*x + 54*x^2 + 108*x^3 + 486*x^4 + 2916*x^5 + 20412*x^6 + ...
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 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( (1 - 12 * x + x * O(x^n))^(3/2), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Nov 25 2013
STATUS
approved