

A213343


1quantum 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
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OFFSET

1,2


COMMENTS

[General discussion]: Consider the 2^N numbers with Ndigit 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,Nq) which is the total number of qquantum 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 ith digit onto quantum states of the ith 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 singlequantum 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, qquantum transitions are detectable by special techniques.
[Specific case]: This sequence is for singlequantum transitions (q = 1). It lists the flattened triangle T(1;N,k), with rows N = 1,2,... and columns k = 0..floor((N1)/2).


REFERENCES

R. R. Ernst, G. Bodenhausen, A. Wokaun, Principles of nuclear magnetic resonance in one and two dimensions, Clarendon Press, 1987, Chapters 26.
M. H. Levitt, Spin Dynamics, J.Wiley & Sons, 2nd Ed.2007, Part3 (Section 6).
J. A. Pople, W. G. Schneider, H. J. Bernstein, Highresolution Nuclear Magnetic Resonance, McGrawHill, 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((N1)/2)
S. Sykora, pQuantum Transitions and a Combinatorial Identity, Stan's Library, II, Aug 2007.


FORMULA

Set q = 1 in: T(q;N,k) = 2^(Nq2*k)*binomial(N,k)*binomial(Nk,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((N1)/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>Nk), return(0), return (binomial(N, k)*binomial(Nk, q+k)*2^((Nk)(q+k)))}
TQ(Nmax, q)={size = 0; for(N=q, nmax, size += 1+(Nq)\2); V = vector(size); idx=0; for(N=q, Nmax, for(k=0, (Nq)\2, idx+=1; V[idx]=TNQK(N, q, k))); return (V); }
WriteVector(file, v, offn)={
for(i=n, #v, write(file, n+offn1, " ", v[n])); }
WriteTq(file, v, noff, q)={idx = 0; n = noff;
while(1, for(k=0, (nq)\2, idx += 1; if(idx>#v, break);
if(k==0, write1(file, n, "  ")); if(k<(nq)\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: A260430 A243347 A317555 * A230057 A205124 A224512
Adjacent sequences: A213340 A213341 A213342 * A213344 A213345 A213346


KEYWORD

nonn,tabf


AUTHOR

Stanislav Sykora, Jun 09 2012


STATUS

approved



