[seqfan] Geometry for A002897, A008977; more Unknowns

[Brad Klee]

In four dimensional geometry, with coordinates {W,X,Y,Z},
there are two nice cuts we can take, with either of the
quadric polynomials:

P1 : ( W^2 + X^2 ) - ( Y^2 + Z^2 ) = 0
P2 : W*Y - X*Z = 0

Both determine toric volumes with polar coordinatizations:

P1: (W,X,Y,Z) = R(C1,S1,C2,S2),
P2: (W,X,Y,Z) = R(C1*S2,C1*C2,S1*C2,S1*S2),
with Sn=sin(xn), Cn=cos(xn).

The second is somehow double covered, because
(x1,x2)-->(x1+Pi,x2+Pi) is an invariant transformation.
If anyone knows a standard name for these coordinate
systems I would like to hear about it (??), but for now
we are just re-inventing this as we go along...

Intersect P1 or P2 with a quartic variety,

P3 : W^2+X^2+Y^2+Z^2 - W*X*Y*Z = t ,

and we find the following surface area integral identities:

A008977 ~ Int_{P1+P3} R^2*dx1*dx2,
A002897 ~ Int_{P2+P3} R^2*dx1*dx2

For some other choice of the quartic perturbation of P3,
the surface area integral may yield A002894 or A000897,
but this is no great surprise. In a wider combinatorial
search, I also found at least two similar sequences (from
a P1 intersection) with relatively slow growth:

a1: 1, 24, 1272, 87360, 6964440, 613698624 . . .
a2: 1, 40, 4056, 569920, 94745560, 17364392640 . . .

I'm not sure these would interest anyone else, but it
only takes about dt=2(s) of computer time to get the
differential equations for the G.F.s. by Griffiths-Dwork
reduction (wow, fast, cool!), so here they are:

576 (27 - 3984 t + 77824 t^2 + 20185088 t^3 + 134217728 t^4) * T
+192 (-10 + 4719 t - 442208 t^2 - 11190272 t^3 + 2153250816 t^4 +
19730006016 t^5) * T'
+12 (1 - 1262 t + 287600 t^2 - 14103552 t^3 - 1276641280 t^4 +
85177925632 t^5 +  996432412672 t^6)*T''
+3 t (11 - 6344 t + 918848 t^2 - 10969088 t^3 - 5254414336 t^4 +
199313326080 t^5 + 2954937499648 t^6)*T^(3)
+t^2 (-1 +  64 t) (-17 + 5288 t - 270848 t^2 - 34471936 t^3 +
2189426688 t^4 + 32212254720 t^5)*T^(4)
+2 t^3 (-1 + 64 t)^2 (-1 + 128 t) (-1 + 24 t + 10240 t^2 + 131072 t^3)*T^(5)
=0

8 (-5 + 1152 t - 1032192 t^2 + 8388608 t^3)*T
+(1 - 2624 t + 1658880 t^2 - 309329920 t^3 + 3556769792 t^4)*T'
+t (11 - 14048 t + 6696960 t^2 - 702021632 t^3 + 8992587776 t^4)*T''
+t^2 (11 - 13184 t + 4472832 t^2 - 341835776 t^3 + 4563402752 t^4)*T^(3)
+2 t^3 (-1 + 64 t) (-1 + 256 t) (1 - 896 t + 16384 t^2)*T^(4)
=0

If nothing else these could be useful as examples or
creative telescoping test cases. Perhaps continued
investigation would reveal more interesting properties?

Cheers --Brad