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%I #6 Jul 06 2023 21:14:24
%S 1,1,2,1,2,12,1,2,67,120,1,2,628,4355,1424,1,2,7779,393128,295234,
%T 19488,1,2,117652,60497283,247268752,21036803,307904,1,2,2097155,
%U 13841757800,470668752866,156500388128,1625419909,5539712,1,2,43046724,4398054899715,1628524328796304,3663682367243907,100264147266880,140823067772,111259904
%N Array read by ascending antidiagonals: A(m, n) = Sum_{i=0..n} Sum_{d=0..n-i} binomial(n, d)*StirlingS2(n-d, i)*(m^(m-1) - 1)^(n-d-i).
%C A(m, n) is the number of all ((m+1)^m)-subgroups of R^n, where R^n is a near-vector space over a proper nearfield R.
%H Prudence Djagba and Jan Hązła, <a href="https://arxiv.org/abs/2306.16421">Combinatorics of subgroups of Beidleman near-vector spaces</a>, arXiv:2306.16421 [math.RA], 2023. See pp. 7-8.
%e The array begins:
%e 1, 2, 12, 120, 1424, 19488, ...
%e 1, 2, 67, 4355, 295234, 21036803, ...
%e 1, 2, 628, 393128, 247268752, 156500388128, ...
%e 1, 2, 7779, 60497283, 470668752866, 3663682367243907, ...
%e ...
%t A[m_,n_]:=Sum[Sum[Binomial[n,d]StirlingS2[n-d,i](m^(m-1)-1)^(n-d-i),{d,0,n-i}],{i,0,n}]; Table[A[m-n+1,n],{m,2,10},{n,0,m-2}]//Flatten
%Y Cf. A003580 (m=2), A364069 (m=3), A364070 (m=4), A364075 (antidiagonal sums).
%K nonn,tabl
%O 2,3
%A _Stefano Spezia_, Jul 04 2023
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