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A213343 1-quantum transitions in systems of N spin 1/2 particles, in columns by combination indices. 10


%S 1,4,12,3,32,24,80,120,10,192,480,120,448,1680,840,35,1024,5376,4480,

%T 560,2304,16128,20160,5040,126,5120,46080,80640,33600,2520,11264,

%U 126720,295680,184800,27720,462

%N 1-quantum transitions in systems of N spin 1/2 particles, in columns by combination indices.

%C [General discussion]: Consider the 2^N numbers with N-digit binary expansion. Let a pair (v,w), here called a "transition", be such that there are exactly k+q digits which are '0' in v and '1' in w, and exactly k digits which are '1' in v and '0' in w. Then T(q;N,k) is the number of all such pairs.

%C For given N and q, the rows of the triangle T(q;N,k) sum up to Sum[k]T(q;N,k) = C(2N,N-q) which is the total number of q-quantum transitions or, equivalently, the number of pairs in which the sum of binary digits of w exceeds that of v by exactly q (see Crossrefs).

%C The terminology stems from the mapping of the i-th digit onto quantum states of the i-th particle (-1/2 for digit '0', +1/2 for digit '1'), the numbers onto quantum states of the system, and the pairs onto quantum transitions between states. In magnetic resonance (NMR) the most intense transitions are the single-quantum ones (q=1) with k=0, called "main transitions", while those with k>0, called "combination transitions", tend to be weaker. Zero-, double- and, in general, q-quantum transitions are detectable by special techniques.

%C [Specific case]: This sequence is for single-quantum transitions (q = 1). It lists the flattened triangle T(1;N,k), with rows N = 1,2,... and columns k = 0..floor((N-1)/2).

%D R. R. Ernst, G. Bodenhausen, A. Wokaun, Principles of nuclear magnetic resonance in one and two dimensions, Clarendon Press, 1987, Chapters 2-6.

%D M. H. Levitt, Spin Dynamics, J.Wiley & Sons, 2nd Ed.2007, Part3 (Section 6).

%D J. A. Pople, W. G. Schneider, H. J. Bernstein, High-resolution Nuclear Magnetic Resonance, McGraw-Hill, 1959, Chapter 6.

%H Stanislav Sykora, <a href="/A213343/b213343.txt">Table of n, a(n) for n = 1..2550</a>

%H Stanislav Sykora, <a href="/A213343/a213343_1.txt">T(1;N,k) with rows N=1,..,100 and columns k=0,..,floor((N-1)/2)</a>

%H S. Sykora, <a href="http://dx.doi.org/10.3247/sl2math07.005">p-Quantum Transitions and a Combinatorial Identity</a>, Stan's Library, II, Aug 2007.

%H Stanislav Sýkora, <a href="http://www.ebyte.it/stan/blog12to14.html#14Dec31">Magnetic Resonance on OEIS</a>, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

%F Set q = 1 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k).

%e T(1;3,1) = 3 because the only transitions compatible with q=1,k=1 are (001,110),(010,101),(100,011).

%e Starting rows of the triangle T(1;N,k):

%e N | k = 0, 1, ..., floor((N-1)/2)

%e 1 | 1

%e 2 | 4

%e 3 | 12 3

%e 4 | 32 24

%e 5 | 80 120 10

%t With[{q = 1}, Table[2^(n - q - 2 k)*Binomial[n, k] Binomial[n - k, q + k], {n, 11}, {k, 0, Floor[(n - 1)/2]}]] // Flatten (* _Michael De Vlieger_, Nov 18 2019 *)

%o (PARI) \\ functions section

%o TNQK(N,q,k)={if ((k<0)||(k>N)||(q+k>N-k),return(0),return (binomial(N,k)*binomial(N-k,q+k)*2^((N-k)-(q+k)))}

%o TQ(Nmax,q)={size = 0;for(N=q,nmax,size += 1+(N-q)\2);V = vector(size);idx=0;for(N=q,Nmax,for(k=0,(N-q)\2,idx+=1;V[idx]=TNQK(N,q,k)));return (V);}

%o WriteVector(file,v,offn)={

%o for(i=n,#v,write(file,n+offn-1," ",v[n]));}

%o WriteTq(file,v,noff,q)={idx = 0;n = noff;

%o while(1,for(k=0,(n-q)\2,idx += 1;if(idx>#v,break);

%o if(k==0,write1(file,n," | "));if(k<(n-q)\2,

%o write1(file,v[idx]," "),write(file,v[idx])));

%o if(idx>#v,break);n += 1;);}

%o \\ execution section:

%o thisq = 1;

%o V = TQ(100,thisq);

%o WriteVector("bfilename",V,thisq);

%o WriteTq("afilename",V,thisq,thisq);

%Y Cf. A051288 (q=0), A213344..A213352 (q=2..10).

%Y Cf. A001787 (first column), A001791 (row sums).

%K nonn,tabf,changed

%O 1,2

%A _Stanislav Sykora_, Jun 09 2012

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Last modified November 20 13:19 EST 2019. Contains 329336 sequences. (Running on oeis4.)