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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).

%I A213343 #51 May 12 2021 10:40:09
%S A213343 1,4,12,3,32,24,80,120,10,192,480,120,448,1680,840,35,1024,5376,4480,
%T A213343 560,2304,16128,20160,5040,126,5120,46080,80640,33600,2520,11264,
%U A213343 126720,295680,184800,27720,462,24576,337920,1013760,887040,221760,11088
%N 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).
%C A213343 [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 A213343 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 A213343 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 A213343 [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 A213343 R. R. Ernst, G. Bodenhausen, A. Wokaun, Principles of nuclear magnetic resonance in one and two dimensions, Clarendon Press, 1987, Chapters 2-6.
%D A213343 M. H. Levitt, Spin Dynamics, J.Wiley & Sons, 2nd Ed.2007, Part3 (Section 6).
%D A213343 J. A. Pople, W. G. Schneider, H. J. Bernstein, High-resolution Nuclear Magnetic Resonance, McGraw-Hill, 1959, Chapter 6.
%H A213343 Stanislav Sykora, <a href="/A213343/b213343.txt">Table of n, a(n) for n = 1..2550</a>
%H A213343 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 A213343 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 A213343 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 A213343 Set q = 1 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k).
%F A213343 T(n, k) = n! * [y^(n-2*k-1)] [x^n] exp(2*x*y)*BesselI(1, 2*x). - _Peter Luschny_, May 12 2021
%e A213343 T(1;3,1) = 3 because the only transitions compatible with q=1,k=1 are (001,110),(010,101),(100,011).
%e A213343 Starting rows of the triangle T(1;N,k):
%e A213343   N | k = 0, 1, ..., floor((N-1)/2)
%e A213343   1 |  1
%e A213343   2 |  4
%e A213343   3 | 12   3
%e A213343   4 | 32  24
%e A213343   5 | 80 120 10
%p A213343 egf := exp(2*x*y) * BesselI(1, 2*x):
%p A213343 ser := series(egf, x, 32): cx := n -> coeff(ser, x, n):
%p A213343 Trow := n -> n!*seq(coeff(cx(n), y, n - 2*k - 1), k = 0..floor((n-1)/2)):
%p A213343 seq(print([n], Trow(n)), n = 1..12); # _Peter Luschny_, May 12 2021
%t A213343 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 A213343 (PARI)
%o A213343 TNQK(N, q, k)={binomial(N, k)*binomial(N-k, q+k)*2^((N-k)-(q+k))}
%o A213343 TQ(Nmax, q)={vector(Nmax-q+1, n, vector(1+(n-1)\2, k, TNQK(n+q-1, q, k-1)))}
%o A213343 { concat(TQ(13, 1)) } \\ simplified by _Andrew Howroyd_, May 12 2021
%Y A213343 Cf. A051288 (q=0), A213344..A213352 (q=2..10).
%Y A213343 Cf. A001787 (first column), A001791 (row sums).
%K A213343 nonn,tabf
%O A213343 1,2
%A A213343 _Stanislav Sykora_, Jun 09 2012