OFFSET
0,9
COMMENTS
Also the number of Krasner factorizations of (x^((p^n*q)^k)-1) / (x-1) into k polynomials each having p^n*q nonzero terms all with coefficient +1. (Krasner and Ranulac, 1937)
LINKS
M. Krasner and B. Ranulac, Sur une propriété des polynomes de la division du cercle, Comptes Rendus Académie des Sciences Paris, 240:397-399, 1937.
Matthew C. Lettington and Karl Michael Schmidt, Divisor Functions and the Number of Sum Systems, arXiv:1910.02455 [math.NT], 2019.
FORMULA
T(n,k) = (n*k)!/((n!)^k*k!) * Sum_{j=0}^k (-n)^(k-j)*binomial(n*k+j,j)*k!/(k-j)!.
T(n,k) = A060540(k,n) * Sum_{j=0}^k (-n)^(k-j)*binomial(n*k+j,j)*k!/(k-j)! for n>=1, k>=1.
EXAMPLE
For two ten-sided dice we have k = 2 and n = 1 since 10 = 2^1*5. The seven configurations are
{{0,1,2,3,4,5,6,7,8,9}, {0,10,20,30,40,50,60,70,80,90}},
{{0,1,2,3,4,50,51,52,53,54}, {0,5,10,15,20,25,30,35,40,45}},
{{0,1,2,3,4,25,26,27,28,29}, {0,5,10,15,20,50,55,60,65,70}},
{{0,1,10,11,20,21,30,31,40,41}, {0,2,4,6,8,50,52,54,56,58}},
{{0,1,20,21,40,41,60,61,80,81}, {0,2,4,6,8,10,12,14,16,18}},
{{0,1,2,3,4,10,11,12,13,14}, {0,5,20,25,40,45,60,65,80,85}},
{{0,1,4,5,8,9,12,13,16,17}, {0,2,20,22,40,42,60,62,80,82}}.
Array begins:
1 1 1 1 1 1 ...
1 1 7 71 1001 18089 ...
1 1 42 3660 614040 169200360 ...
1 1 230 160440 299145000 1175153779800 ...
1 1 1190 6387150 127534407000 6888547183518000 ...
1 1 5922 238504266 49829456981304 36179571823974699120 ...
1 1 28644 8507955456 18306027156441024 175934152220744900062080 ...
...
PROG
(SageMath)
def T(n, k):
return(factorial(k*n)/factorial(n)^k/factorial(k)\
* sum((-n)^(k-j)*binomial(n*k+j, j)*falling_factorial(k, j)\
for j in range(k+1)))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
William P. Orrick, Feb 18 2023
STATUS
approved