OFFSET
0,3
COMMENTS
In Redfield 1927 on page 443 he writes "If in V we put 1/(1-x^r) for every s_r, we obtain the infinite series 1 + x + 4x^2 + 7x^3 + 21x^4 + 37x^5 + ..., in which the coefficient of x^t enumerates the distinct configurations obtained by placing a zero or a positive integer at every vertex of the cube, subject to the condition that the sum of the 8 numbers is always t.". - Michael Somos, Oct 17 2015
Note that the enumeration is modded out by the symmetries of the cube. - Michael Somos, Oct 17 2015
REFERENCES
J. H. Redfield, The theory of group-reduced distributions, Amer. J. Math., 49 (1927), 433-455; reprinted in P. A. MacMahon, Coll. Papers I, pp. 805-827.
FORMULA
G.f.: (x^12 - x^11 + x^10 + 6*x^8 + x^7 + 8*x^6 + x^5 + 6*x^4 + x^2 - x + 1) / ((1 - x) * (1 - x^2) * (1 - x^3) * (1 - x^4))^2. - Michael Somos, Mar 05 2004
G.f.: (1/24) * (1 - x)^-8 + (3/8) * (1 - x^2)^-4 + (1/3) * (1 - x)^-2 * (1 - x^3)^-2 + (1/4) * (1 - x^4)^-2. - Michael Somos, Oct 17 2015
a(n) = -a(-8 - n) for all n in Z. - Michael Somos, Oct 17 2015
EXAMPLE
For n=2 the 4 ways are: {0000 0002}, {0000 0011}, {0001 0100}, {0001 1000}.
G.f. = 1 + x + 4*x^2 + 7*x^3 + 21*x^4 + 37*x^5 + 85*x^6 + 151*x^7 + 292*x^8 + ...
MAPLE
1/24/(1-x)^8+3/8/(1-x^2)^4+1/3/(1-x^3)^2/(1-x)^2+1/4/(1-x^4)^2;
MATHEMATICA
a[ n_] := Ceiling[ (3 n^7 + 84 n^6 + 966 n^5 + 5880 n^4 + If[ OddQ@n, 22547 n^3 + 66276 n^2, 25382 n^3 + 100296 n^2] + 12 n (10547 + 35 If[ OddQ@n, If[ Mod[n, 6] < 5, 32, 0], If[ Mod[n, 6] == 2, 297, 329] + 54 Boole[Mod[n, 4] == 0]]) + 1) / 362880]; (* Michael Somos, Oct 17 2015 *)
PROG
(PARI) {a(n) = if( n<-4, -a(-8 - n), polcoeff( subst( Pol([ 1, -1, -5, 5, 11, -4, -4]), x, x + 1/x) * x^6 / prod(k=1, 4, 1 - x^k)^2 + x * O(x^n), n))}; /* Michael Somos, Mar 05 2004 */
(PARI) {a(n) = ceil( (3*n^7 + 84*n^6 + 966*n^5 + 5880*n^4 + if( n%2, 22547*n^3 + 66276*n^2, 25382*n^3 + 100296*n^2) + 12*n * (10547 + 35 * if( n%2, if( n%6<5, 32, 0), if( n%6==2, 297, 329) + 54*(n%4==0))) + 1) / 362880)}; /* Michael Somos, Oct 17 2015 */
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved