A039959 Number of ways of numbering the vertices of a cube so sum of the 8 numbers is n.

1, 1, 4, 7, 21, 37, 85, 151, 292, 490, 848, 1346, 2157, 3260, 4925, 7148, 10327, 14477, 20177, 27483, 37194, 49431, 65277, 84945, 109873, 140394, 178377, 224334, 280647, 348040, 429526, 526108, 641524, 777127, 937513, 1124461, 1343567, 1597115, 1891850, 2230685, 2621671, 3068438

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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

Examples

			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 + ...
		

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.

Links

• Ray Chandler, Table of n, a(n) for n = 0..100
• Index entries for two-way infinite sequences
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1,-4,4,4,2,0,-10,0,2,4,4,-4,-1,-2,1,2,-1).

Programs

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 *)

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 */

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