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

Author

William P. Orrick, Feb 18 2023

Keywords

• nonn
• tabl

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)

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

For a table with the number of sides not restricted to the form p^n*q see A360098.

T(n,2) = A349427(n+1).

T(1,k) = |A002119(k)|.

Cf. A131514, A273013.

Programs

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.