OFFSET
1,2
COMMENTS
Each member of a chiral pair is a reflection, but not a rotation, of the other. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual.
LINKS
Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).
FORMULA
a(n) = (n-1) * n^2 * (n+1) * (n^2+2) * (n^14 - n^12 + 3*n^10 - 5*n^8 - 4*n^6 + 8*n^4 + 4*n^2 + 12) /120.
a(n) = 8388*C(n,2) + 28972926*C(n,3) + 9044690976*C(n,4) + 749186015850*C(n,5) + 25836356193012*C(n,6) + 468028878138864*C(n,7) + 5097432576698784*C(n,8) + 36322117709159520*C(n,9) + 178947768558202560*C(n,10) + 632296225414909440*C(n,11) + 1640646875114311680*C(n,12) + 3168965153453299200*C(n,13) + 4578694359419980800*C(n,14) + 4929160839482880000*C(n,15) + 3897035952819609600*C(n,16) + 2197214626134528000*C(n,17) + 836310065310720000*C(n,18) + 192604742313984000*C(n,19) + 20274183401472000*C(n,20), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
MATHEMATICA
Table[(n^20-15n^12+14n^10+20n^8+4n^4-24n^2)/120, {n, 30}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Oct 03 2020
STATUS
approved