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A213343 1-quantum transitions in systems of N spin 1/2 particles, in columns by combination indices. 10
1, 4, 12, 3, 32, 24, 80, 120, 10, 192, 480, 120, 448, 1680, 840, 35, 1024, 5376, 4480, 560, 2304, 16128, 20160, 5040, 126, 5120, 46080, 80640, 33600, 2520, 11264, 126720, 295680, 184800, 27720, 462 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

[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.

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).

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.

[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).

REFERENCES

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

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

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

LINKS

Stanislav Sykora, Table of n, a(n) for n = 1..2550

Stanislav Sykora, T(1;N,k) with rows N=1,..,100 and columns k=0,..,floor((N-1)/2)

S. Sykora, p-Quantum Transitions and a Combinatorial Identity, Stan's Library, II, Aug 2007.

FORMULA

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

EXAMPLE

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

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

  N | k = 0, 1, ..., floor((N-1)/2)

  1 |  1

  2 |  4

  3 | 12   3

  4 | 32  24

  5 | 80 120 10

PROG

(PARI) \\ functions section

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)))}

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); }

WriteVector(file, v, offn)={

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

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

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

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

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

    if(idx>#v, break); n += 1; ); }

\\ execution section:

thisq = 1;

V = TQ(100, thisq);

WriteVector("bfilename", V, thisq);

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

CROSSREFS

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

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

Sequence in context: A104063 A260430 A243347 * A230057 A205124 A224512

Adjacent sequences:  A213340 A213341 A213342 * A213344 A213345 A213346

KEYWORD

nonn,tabf

AUTHOR

Stanislav Sykora, Jun 09 2012

STATUS

approved

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Last modified May 22 16:19 EDT 2017. Contains 286882 sequences.