cp's OEIS Frontend

A165410 Hankel transform of the transform of 2^n given by A165409.

%I A165410 #11 Apr 29 2023 20:47:06
%S A165410 1,0,-4,-16,0,1024,16384,0,-16777216,-1073741824,0,17592186044416,
%T A165410 4503599627370496,0,-1180591620717411303424,
%U A165410 -1208925819614629174706176,0,5070602400912917605986812821504,20769187434139310514121985316880384
%N A165410 Hankel transform of the transform of 2^n given by A165409.
%C A165410 Powers of two occurring in this sequence are based on the hexagonal spiral pattern of A049450 and A049451 (see A152749):
%C A165410 .
%C A165410         16--15--14
%C A165410         /         \
%C A165410       17   5---4  13
%C A165410       /   /     \   \
%C A165410     18   6   0   3  12
%C A165410     /   /   /   /   /
%C A165410   19   7   1---2  11  26
%C A165410     \   \         /   /
%C A165410     20   8---9--10  25
%C A165410       \             /
%C A165410       21--22--23--24
%C A165410 .
%C A165410 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 A165410 The Hankel transforms of transforms of r^n behave similarly -- we get 1, 0, -r^2, -r^4, 0, r^10, r^14, ....
%C A165410 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 A165410 Cf. A165409.
%K A165410 sign
%O A165410 0,3
%A A165410 _Paul Barry_, Sep 17 2009