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%I #11 Apr 29 2023 20:47:06
%S 1,0,-4,-16,0,1024,16384,0,-16777216,-1073741824,0,17592186044416,
%T 4503599627370496,0,-1180591620717411303424,
%U -1208925819614629174706176,0,5070602400912917605986812821504,20769187434139310514121985316880384
%N Hankel transform of the transform of 2^n given by A165409.
%C Powers of two occurring in this sequence are based on the hexagonal spiral pattern of A049450 and A049451 (see A152749):
%C .
%C 16--15--14
%C / \
%C 17 5---4 13
%C / / \ \
%C 18 6 0 3 12
%C / / / / /
%C 19 7 1---2 11 26
%C \ \ / /
%C 20 8---9--10 25
%C \ /
%C 21--22--23--24
%C .
%C The powers, (0,-oo,2,4,-oo,10,14,-oo,24,30,-oo,...) correspond to vertically joining pairs on the (0,4) and (0,2) radial lines, with -oo corresponding to the jump to the next pair.
%C The Hankel transforms of transforms of r^n behave similarly -- we get 1, 0, -r^2, -r^4, 0, r^10, r^14, ....
%C Note the Somos-4 property: a(3n) = 4*a(3n-1)*a(3n-3)/e(3n-4). Related to elliptic curve y^2 = 1 - 8x^3 in g.f. of A165409.
%Y Cf. A165409.
%K sign
%O 0,3
%A _Paul Barry_, Sep 17 2009
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