A292881
Number of n-step closed paths on the E6 lattice.
Original entry on oeis.org
1, 0, 72, 1440, 54216, 2134080, 93993120, 4423628160, 219463602120, 11341793393280
Offset: 0
The 2-step walks consist of hopping to one of the 72 minimal vectors of the E6 lattice and then back to the origin.
A292882
Number of n-step closed paths on the E7 lattice.
Original entry on oeis.org
1, 0, 126, 4032, 228690, 14394240, 1020623940, 78353170560, 6393827197170
Offset: 0
The 2-step walks consist of hopping to one of the 126 minimal vectors of the E7 lattice and then back to the origin.
A292883
Number of n-step closed paths on the E8 lattice.
Original entry on oeis.org
1, 0, 240, 13440, 1260720, 137813760, 17141798400, 2336327078400, 341350907713200
Offset: 0
The 2-step walks consist of hopping to one of the 240 minimal vectors of the E8 lattice and then back to the origin.
A328735
Constant term in the expansion of (x + y + z + 1/x + 1/y + 1/z + x*y + y*z + z*x + 1/(x*y) + 1/(y*z) + 1/(z*x) + x*y*z + 1/(x*y*z))^n.
Original entry on oeis.org
1, 0, 14, 72, 882, 8400, 95180, 1060080, 12389650, 146472480, 1767391164, 21581516880, 266718438756, 3327025429728, 41849031952728, 530135326392672, 6757845419895570, 86619827323917888, 1115719258312182524, 14434274832755201424, 187477238295444829732
Offset: 0
Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^m:
A126869 (m=2),
A002898 (m=3), this sequence (m=4),
A328751 (m=5).
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Table[Sum[(-2)^(n-i)*Binomial[n,i] * Sum[Binomial[i,j]^4, {j,0,i}], {i,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 20 2023 *)
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{a(n) = polcoef(polcoef(polcoef((-2+(1+x)*(1+y)*(1+z)+(1+1/x)*(1+1/y)*(1+1/z))^n, 0), 0), 0)}
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{a(n) = sum(i=0, n, (-2)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^4))}
A342800
Number of self-avoiding polygons on a 3-dimensional cubic lattice where each walk consists of steps with incrementing length from 1 to n.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 24, 72, 0, 0, 1704, 5184, 0, 0, 193344, 600504, 0, 0, 34321512, 141520752, 0, 0, 9205815672, 37962945288, 0, 0
Offset: 1
a(1) to a(6) = 0 as no self-avoiding closed-loop walk is possible.
a(7) = 24 as there is one walk which forms a closed loop which can be walked in 24 different ways on a 3D cubic lattice. These walks, and those for n(8) = 72, are purely 2-dimensional. See A334720 for images of these walks.
a(11) = 1704. These walks consist of 120 purely 2-dimensional walks and 1584 3-dimensional walks. One of these 3-dimensional walks is:
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/|
/ | z y
/ | | /
7 +y / | |/
/ | 8 -z |----- x
6 +x / |
|---.---.---.---.---.---/ | 9 +x
| |---.---.---.---.---.---.---.---.---/
| 5 +z /
| /
|---.---.---.---/ /
4 -x / 3 +y /
/ / 10 -y
| 2 +z /
| /
| 1 +z /
X---.---.---.---.---.---.---.---.---.---.---/
11 -x
.
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