|
%I #11 Feb 10 2024 13:42:27
%S 2,2,6,1,5,16,1,2,12,34,1,1,6,28,77,1,1,2,14,61,157,1,1,1,6,34,133,
%T 328,1,1,1,2,15,74,277,655,1,1,1,1,6,36,165,572,1309,1,1,1,1,2,15,80,
%U 345,1154,2550,1,1,1,1,1,6,37,179,722,2299,4958
%N Triangle read by rows: number of inequivalent classes of true tripartite entanglement systems of type 2 X M X N, for 2 <= N <= M.
%H XiKun Li, JunLi Li, Bin Liu and CongFeng Qiao, <a href="https://doi.org/10.1007/s11433-011-4395-9">The parametric symmetry and numbers of the entangled class of 2 × M × N system</a>, SCIENCE CHINA PHYSICS, MECHANICS & ASTRONOMY, Volume 54, Number 8, 1471-1475
%e Triangle begins:
%e 2
%e 2 6
%e 1 5 16
%e 1 2 12 34
%e 1 1 6 28 77
%e 1 1 2 14 61 157
%e 1 1 1 6 34 133 328
%e 1 1 1 2 15 74 277 655
%e 1 1 1 1 6 36 165 572 1309
%e ...
%e This may also be regarded as a square array, allowing values on N >= M, in which case it begins like this:
%e 2 2 1 1 1 1 1 1 1 ...
%e 2 6 5 2 1 1 1 1 1 ...
%e 1 5 16 12 6 2 1 1 1 ...
%e 1 2 12 34 28 14 6 2 1 ...
%e 1 1 6 28 77 61 34 15 6 ...
%e 1 1 2 14 61 157 133 74 36 ...
%e 1 1 1 6 34 133 328 277 165 ...
%e 1 1 1 2 15 74 277 655 572 ...
%e 1 1 1 1 6 36 165 572 1309 ...
%e ...
%p f := proc(n,m)
%p 1/mul(1-x^k,k=1..m) ;
%p coeftayl(%,x=0,n) ;
%p end proc:
%p F := proc(j,r,c)
%p option remember ;
%p if j < 0 then
%p return 0 ;
%p end if;
%p if j = 0 then
%p 1 ;
%p elif c = 0 then
%p f(j,r) ;
%p elif r=0 then
%p f(j,c) ;
%p else
%p procname(j,r,0)+procname(j,0,c) + add(add(procname(j-m-n,m,n),n=1..c),m=1..r) ;
%p end if ;
%p end proc:
%p omega := proc(M,N,i,j)
%p option remember;
%p A001970(2*M-N-3*i-j)*F(j,i,i+N-M) ;
%p end proc:
%p A199476 := proc(M,N)
%p local a,i,j ;
%p if N< M then
%p return procname(N,M);
%p end if;
%p if N >= 2*M then
%p return 1 ;
%p end if;
%p a := 0 ;
%p for i from 0 to floor((2*M-N)/3) do
%p for j from 0 to 2*M-N-3*i do
%p a := a+ omega(M,N,i,j) ;
%p if M = N and i =0 and j =0 then
%p a := a-1 ;
%p end if;
%p end do:
%p end do:
%p a ;
%p end proc:
%p seq( seq(A199476(N,M),N=2..M),M=2..12) ; # _R. J. Mathar_, Feb 10 2024
%Y Cf. A199477.
%K nonn,tabl
%O 2,1
%A _N. J. A. Sloane_, Nov 06 2011
|
