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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.
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%I #9 Mar 24 2023 16:02:26

%S 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,

%T 614040,18089,1,1,1,5922,6387150,299145000,169200360,398959,1,1,1,

%U 28644,238504266,127534407000,1175153779800,69444920160,10391023,1

%N 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.

%C 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)

%H M. Krasner and B. Ranulac, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k31562/f397.item">Sur une propriété des polynomes de la division du cercle</a>, Comptes Rendus Académie des Sciences Paris, 240:397-399, 1937.

%H Matthew C. Lettington and Karl Michael Schmidt, <a href="https://arxiv.org/abs/1910.02455">Divisor Functions and the Number of Sum Systems</a>, arXiv:1910.02455 [math.NT], 2019.

%F T(n,k) = (n*k)!/((n!)^k*k!) * Sum_{j=0}^k (-n)^(k-j)*binomial(n*k+j,j)*k!/(k-j)!.

%F 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.

%e For two ten-sided dice we have k = 2 and n = 1 since 10 = 2^1*5. The seven configurations are

%e {{0,1,2,3,4,5,6,7,8,9}, {0,10,20,30,40,50,60,70,80,90}},

%e {{0,1,2,3,4,50,51,52,53,54}, {0,5,10,15,20,25,30,35,40,45}},

%e {{0,1,2,3,4,25,26,27,28,29}, {0,5,10,15,20,50,55,60,65,70}},

%e {{0,1,10,11,20,21,30,31,40,41}, {0,2,4,6,8,50,52,54,56,58}},

%e {{0,1,20,21,40,41,60,61,80,81}, {0,2,4,6,8,10,12,14,16,18}},

%e {{0,1,2,3,4,10,11,12,13,14}, {0,5,20,25,40,45,60,65,80,85}},

%e {{0,1,4,5,8,9,12,13,16,17}, {0,2,20,22,40,42,60,62,80,82}}.

%e Array begins:

%e 1 1 1 1 1 1 ...

%e 1 1 7 71 1001 18089 ...

%e 1 1 42 3660 614040 169200360 ...

%e 1 1 230 160440 299145000 1175153779800 ...

%e 1 1 1190 6387150 127534407000 6888547183518000 ...

%e 1 1 5922 238504266 49829456981304 36179571823974699120 ...

%e 1 1 28644 8507955456 18306027156441024 175934152220744900062080 ...

%e ...

%o (SageMath)

%o def T(n,k):

%o return(factorial(k*n)/factorial(n)^k/factorial(k)\

%o * sum((-n)^(k-j)*binomial(n*k+j,j)*falling_factorial(k,j)\

%o for j in range(k+1)))

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

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

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

%Y Cf. A131514, A273013.

%K nonn,tabl

%O 0,9

%A _William P. Orrick_, Feb 18 2023