cp's OEIS Frontend

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.

%I A360439 #9 Mar 24 2023 16:02:26
%S A360439 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 A360439 614040,18089,1,1,1,5922,6387150,299145000,169200360,398959,1,1,1,
%U A360439 28644,238504266,127534407000,1175153779800,69444920160,10391023,1
%N 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.
%C A360439 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 A360439 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 A360439 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 A360439 T(n,k) = (n*k)!/((n!)^k*k!) * Sum_{j=0}^k (-n)^(k-j)*binomial(n*k+j,j)*k!/(k-j)!.
%F A360439 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 A360439 For two ten-sided dice we have k = 2 and n = 1 since 10 = 2^1*5. The seven configurations are
%e A360439   {{0,1,2,3,4,5,6,7,8,9}, {0,10,20,30,40,50,60,70,80,90}},
%e A360439   {{0,1,2,3,4,50,51,52,53,54}, {0,5,10,15,20,25,30,35,40,45}},
%e A360439   {{0,1,2,3,4,25,26,27,28,29}, {0,5,10,15,20,50,55,60,65,70}},
%e A360439   {{0,1,10,11,20,21,30,31,40,41}, {0,2,4,6,8,50,52,54,56,58}},
%e A360439   {{0,1,20,21,40,41,60,61,80,81}, {0,2,4,6,8,10,12,14,16,18}},
%e A360439   {{0,1,2,3,4,10,11,12,13,14}, {0,5,20,25,40,45,60,65,80,85}},
%e A360439   {{0,1,4,5,8,9,12,13,16,17}, {0,2,20,22,40,42,60,62,80,82}}.
%e A360439 Array begins:
%e A360439   1  1      1           1                  1                         1  ...
%e A360439   1  1      7          71               1001                     18089  ...
%e A360439   1  1     42        3660             614040                 169200360  ...
%e A360439   1  1    230      160440          299145000             1175153779800  ...
%e A360439   1  1   1190     6387150       127534407000          6888547183518000  ...
%e A360439   1  1   5922   238504266     49829456981304      36179571823974699120  ...
%e A360439   1  1  28644  8507955456  18306027156441024  175934152220744900062080  ...
%e A360439   ...
%o A360439 (SageMath)
%o A360439 def T(n,k):
%o A360439     return(factorial(k*n)/factorial(n)^k/factorial(k)\
%o A360439      * sum((-n)^(k-j)*binomial(n*k+j,j)*falling_factorial(k,j)\
%o A360439      for j in range(k+1)))
%Y A360439 For a table with the number of sides not restricted to the form p^n*q see A360098.
%Y A360439 T(n,2) = A349427(n+1).
%Y A360439 T(1,k) = |A002119(k)|.
%Y A360439 Cf. A131514, A273013.
%K A360439 nonn,tabl
%O A360439 0,9
%A A360439 _William P. Orrick_, Feb 18 2023