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

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.

Links

Table of n, a(n) for n=0..38.

Index entries for two-way infinite sequences

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

Sequence in context: A368185 A359603 A255512 * A320663 A186335 A010363

Adjacent sequences: A039956 A039957 A039958 * A039960 A039961 A039962

Keyword

nonn

Author

N. J. A. Sloane

Status

approved