A360439 Square array read by antidiagonals upwards: T(n,k), n>=0, k>=0, is the number of ways of choosing nonnegative numbers for k indistinguishable (p^n*q)-sided dice so that it is possible to roll every number from 0 to (p^n*q)^k-1, where p and q are distinct primes.
1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 42, 71, 1, 1, 1, 230, 3660, 1001, 1, 1, 1, 1190, 160440, 614040, 18089, 1, 1, 1, 5922, 6387150, 299145000, 169200360, 398959, 1, 1, 1, 28644, 238504266, 127534407000, 1175153779800, 69444920160, 10391023, 1
Offset: 0
Examples
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 ...
...
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.
Crossrefs
Programs
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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)))
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.
Comments