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

1, 0, -4, -16, 0, 1024, 16384, 0, -16777216, -1073741824, 0, 17592186044416, 4503599627370496, 0, -1180591620717411303424, -1208925819614629174706176, 0, 5070602400912917605986812821504, 20769187434139310514121985316880384

Offset: 0

Author

Paul Barry, Sep 17 2009

Keywords

• sign

Comments

Powers of two occurring in this sequence are based on the hexagonal spiral pattern of A049450 and A049451 (see A152749):

.

16--15--14

/ \

17 5---4 13

/ / \ \

18 6 0 3 12

/ / / / /

19 7 1---2 11 26

\ \ / /

20 8---9--10 25

\ /

21--22--23--24

.

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.

The Hankel transforms of transforms of r^n behave similarly -- we get 1, 0, -r^2, -r^4, 0, r^10, r^14, ....

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.

Crossrefs

Cf. A165409.