A213343 1-quantum transitions in systems of N spin 1/2 particles, in columns by combination indices. Triangle read by rows, T(n, k) for n >= 1 and 0 <= k <= floor((n-1)/2).

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, 24576, 337920, 1013760, 887040, 221760, 11088

Offset: 1

Author

Stanislav Sykora, Jun 09 2012

Keywords

• nonn
• tabf

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).

Examples

			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
		

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.
• Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Crossrefs

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

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

Programs

Maple

egf := exp(2*x*y) * BesselI(1, 2*x):
ser := series(egf, x, 32): cx := n -> coeff(ser, x, n):
Trow := n -> n!*seq(coeff(cx(n), y, n - 2*k - 1), k = 0..floor((n-1)/2)):
seq(print([n], Trow(n)), n = 1..12); # Peter Luschny, May 12 2021

Mathematica

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 *)

PARI

TNQK(N, q, k)={binomial(N, k)*binomial(N-k, q+k)*2^((N-k)-(q+k))}
TQ(Nmax, q)={vector(Nmax-q+1, n, vector(1+(n-1)\2, k, TNQK(n+q-1, q, k-1)))}
{ concat(TQ(13, 1)) } \\ simplified by Andrew Howroyd, May 12 2021

Formula

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

T(n, k) = n! * [y^(n-2*k-1)] [x^n] exp(2*x*y)*BesselI(1, 2*x). - Peter Luschny, May 12 2021