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A370887
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.
0
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, 2664, 2825, 1, 2, 16, 268, 3652, 42176, 56632, 29212, 1, 2, 20, 368, 19156, 285704, 3583232, 2052656, 417199, 1, 2, 22, 616, 35872, 3961832, 61946920, 666124288
OFFSET
0,3
COMMENTS
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.
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).
FORMULA
T(n,k) = 2*T(n-1,k) + (A000040(k)^(n-1)-1)*T(n-2,k).
T(0,k) = 1.
T(1,k) = 2.
T(2,k) = A000040(k) + 3 = A113935(k).
T(3,k) = 2*(A000040(k)^3 + (A000040(k)-2))/(A000040(k)-1).
EXAMPLE
T(1,1) = 2 since the elementary abelian of order A000040(1)^1 = 2^1 has 2 subgroups.
T(3,5) = 2*T(2,5) + (A000040(5)^(3-1)-1)*T(1,5) = 2*14 + ((11^2)-1)*2 = 268.
First 6 rows and 8 columns:
n\k| 1 2 3 4 5 6 7 8
----+---------------------------------------------------------------------------
0 | 1 1 1 1 1 1 1 1
1 | 2 2 2 2 2 2 2 2
2 | 5 6 8 10 14 16 20 22
3 | 16 28 64 116 268 368 616 764
4 | 67 212 1120 3652 19156 35872 99472 152404
5 | 374 2664 42176 285704 3961832 10581824 51647264 99869288
6 |2825 56632 3583232 61946920 3092997464 13340150272 141339210560 377566978168
PROG
(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)
(GAP)
# produces an array A of the first (7(7+1))/2 terms. However computation quickly becomes expensive for values > 7.
LoadPackage("sonata"); # sonata package needs to be loaded to call function Subgroups. Sonata is included in latest versions of GAP.
N:=[1..7];; R:=[];; S:=[];;
for i in N do
for j in N do
if j>i then
break;
fi;
Add(R, j);
od;
Add(S, R);
R:=[];;
od;
A:=[];;
for n in N do
L:=List([1..Length(S[n])], m->Size(Subgroups(ElementaryAbelianGroup( Primes[Reversed(S[n])[m]]^(S[n][m]-1)))));
Add(A, L);
od;
A:=Flat(A);
KEYWORD
nonn,tabl
AUTHOR
Miles Englezou, Mar 05 2024
STATUS
approved