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
Examples
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
Programs
-
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); -
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)
Comments