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%I #35 Mar 29 2024 05:43:01
%S 1,1,2,1,2,5,1,2,6,16,1,2,8,28,67,1,2,10,64,212,374,1,2,14,116,1120,
%T 2664,2825,1,2,16,268,3652,42176,56632,29212,1,2,20,368,19156,285704,
%U 3583232,2052656,417199,1,2,22,616,35872,3961832,61946920,666124288
%N Square array read by descending antidiagonals: T(n,k) is the number of subgroups of the elementary abelian group of order A000040(k)^n for n >= 0 and k >= 1.
%C As an elementary abelian group G of order p^n is isomorphic to an n-dimensional vector space V over the finite field of characteristic p, T(n,k) is also the number of subspaces of V.
%C V defined as above, T(n,k) is also the sum of the Gaussian binomial coefficients (n,r), 0 <= r < n, for a prime q number, since (n,r) counts the number of r-dimensional subspaces of V. The sequences of these sums for a fixed prime q number correspond to the columns of T(n,k).
%F T(n,k) = 2*T(n-1,k) + (A000040(k)^(n-1)-1)*T(n-2,k).
%F T(0,k) = 1.
%F T(1,k) = 2.
%F T(2,k) = A000040(k) + 3 = A113935(k).
%F T(3,k) = 2*(A000040(k)^3 + (A000040(k)-2))/(A000040(k)-1).
%e T(1,1) = 2 since the elementary abelian of order A000040(1)^1 = 2^1 has 2 subgroups.
%e T(3,5) = 2*T(2,5) + (A000040(5)^(3-1)-1)*T(1,5) = 2*14 + ((11^2)-1)*2 = 268.
%e First 6 rows and 8 columns:
%e n\k| 1 2 3 4 5 6 7 8
%e ----+---------------------------------------------------------------------------
%e 0 | 1 1 1 1 1 1 1 1
%e 1 | 2 2 2 2 2 2 2 2
%e 2 | 5 6 8 10 14 16 20 22
%e 3 | 16 28 64 116 268 368 616 764
%e 4 | 67 212 1120 3652 19156 35872 99472 152404
%e 5 | 374 2664 42176 285704 3961832 10581824 51647264 99869288
%e 6 |2825 56632 3583232 61946920 3092997464 13340150272 141339210560 377566978168
%o (PARI) T(n,k)=polcoeff(sum(i=0,n,x^i/prod(j=0,i,1-primes(k)[k]^j*x+x*O(x^n))),n)
%o (GAP)
%o # produces an array A of the first (7(7+1))/2 terms. However computation quickly becomes expensive for values > 7.
%o LoadPackage("sonata"); # sonata package needs to be loaded to call function Subgroups. Sonata is included in latest versions of GAP.
%o N:=[1..7];; R:=[];; S:=[];;
%o for i in N do
%o for j in N do
%o if j>i then
%o break;
%o fi;
%o Add(R,j);
%o od;
%o Add(S,R);
%o R:=[];;
%o od;
%o A:=[];;
%o for n in N do
%o L:=List([1..Length(S[n])],m->Size(Subgroups(ElementaryAbelianGroup( Primes[Reversed(S[n])[m]]^(S[n][m]-1)))));
%o Add(A,L);
%o od;
%o A:=Flat(A);
%Y Cf. A000040, A113935, A006116, A006117, A006119, A006121, A015197.
%K nonn,tabl
%O 0,3
%A _Miles Englezou_, Mar 05 2024
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