A213344 -id:A213344 - cp's OEIS frontend

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

Stanislav Sykora, Jun 09 2012

[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).

			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

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.

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.

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

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

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

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 *)

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

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

A213345 3-quantum transitions in systems of N>=3 spin 1/2 particles, in columns by combination indices.

Original entry on oeis.org

1, 8, 40, 5, 160, 60, 560, 420, 21, 1792, 2240, 336, 5376, 10080, 3024, 84, 15360, 40320, 20160, 1680, 42240, 147840, 110880, 18480, 330, 112640, 506880, 532224, 147840, 7920, 292864, 1647360, 2306304, 960960, 102960

Offset: 3

Stanislav Sykora, Jun 12 2012

For a general discussion, please see A213343.
This a(n) is for triple-quantum transitions (q = 3).
It lists the flattened triangle T(3;N,k) with rows N = 3,5,... and columns k = 0..floor((N-3)/2).

			Some of the 40 triple-quantum transitions for N = 5 and combination index 0: (00000,01011),(10010,11111),...
Starting rows of the triangle T(3;N,k):
  N | k = 0, 1, ..., floor((N-3)/2)
  3 |   1
  4 |   8
  5 |  40   5
  6 | 160  60
  7 | 560 420 21

See A213343

Stanislav Sykora, Table of n, a(n) for n = 3..2452
Stanislav Sykora, T(3;N,k) with rows N=3,..,100 and columns k=0,..,floor((N-3)/2)
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Cf. A051288 (q=0), A213343 (q=1), A213344 (q=2), A213346 to A213352 (q=4..10).

Cf. A001789 (first column), A002696 (row sums).

With[{q = 3}, Table[2^(n - q - 2 k)*Binomial[n, k] Binomial[n - k, q + k], {n, 13}, {k, 0, Floor[(n - q)/2]}]] // Flatten (* Michael De Vlieger, Nov 18 2019 *)

PARI
```
See A213343; set thisq = 3.
```

Set q = 3 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k).

cp's OEIS Frontend

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).

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