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.

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).

Links

Table of n, a(n) for n=0..52.

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);

Crossrefs

Cf. A000040, A113935, A006116, A006117, A006119, A006121, A015197.

Sequence in context: A122881 A210217 A334165 * A210223 A361600 A242598

Adjacent sequences: A370884 A370885 A370886 * A370888 A370889 A370890

Keyword

nonn,tabl

Author

Miles Englezou, Mar 05 2024

Status

approved